Dmitrii Kouznetsov

Superfunctions
Non-integer iterates of holomorphic functions.
Tetration and other superfunctions. Formulas,algorithms, tables, graphics and complex maps.

2020

Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020

https://www.morebooks.shop/bookprice_offer_ee30ef117c605973c58d4db47a4a01251acab641

https://mizugadro.mydns.jp/BOOK/444.pdf or
https://mizugadro.mydns.jp/BOOK/465.pdf or
https://mizugadro.mydns.jp/BOOK/466.pdf (or even bigger number)This version is generated from the Latex source July 29, 2020
The Russian version of year 2014 (Fig.1), Isbn: 978-3-659-56202-0

https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0

List of misprints in the First edition of the Russian version is athttps://mizugadro.mydns.jp/t/index.php/Superfunctions_in_Russian
I load the corrected Russian version as

https://mizugadro.mydns.jp/BOOK/202.pdf
https://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
List of figures (for both English and versions):

https://mizugadro.mydns.jp/t/index.php/Category:Book

Fig. 1: Distribution of the Russian version of this Book, 2014

Contents
1

Prologue

1
2
3
4
5
6
7
2

For whom is this Prologue . . . . . . . . . . . . . . . .
Russian version and copyleft . . . . . . . . . . . . . . .
Physical mathematics . . . . . . . . . . . . . . . . . . .
TORI axioms . . . . . . . . . . . . . . . . . . . . . . .
What for this book is . . . . . . . . . . . . . . . . . . .
What is present in the Book . . . . . . . . . . . . . . .
What is absent in the Book . . . . . . . . . . . . . . . .
Introduction

1
2
3
4
5
6
7
8
3

4

Use of superscript . . . . . . . . . . . . . . . . . . . . .
Transfer equation . . . . . . . . . . . . . . . . . . . . .
Values of the argument . . . . . . . . . . . . . . . . . .
Nest . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fibre amplifier . . . . . . . . . . . . . . . . . . . . . . .
Transfer equation and the Abel equation . . . . . . . . .
Multiplicity of solutions . . . . . . . . . . . . . . . . . .
Formulation of the problem . . . . . . . . . . . . . . . .

21
22
23
24
24
25
26
28
29

Old trick . . . . . . . . . . . . . . . . . . . . . . . . . .
Construction of elementary superfunctions . . . . . . . .
Elementary superfunctions

1
2
3

8
9
11
11
13
16
17
20

Table of superfunctions

1
2

5

7

Iteration of linear function . . . . . . . . . . . . . . . .
Rational function . . . . . . . . . . . . . . . . . . . . .
Power function and its iterates . . . . . . . . . . . . . .

29
32
33

33
35
43

Tania and Shoka

46

1
2
3
4

47
51
53
60

Tania and Arctania . . . . . . . . . . . . . . . . . . . .
Transfer function Doya . . . . . . . . . . . . . . . . . .
Keller, Shoka and ArcShoka . . . . . . . . . . . . . . . .
Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
3

6

Regular iteration

1
2
3
4
7

General formula . . . . . . . . . . . . . . . . . . . . . .
Exact solution . . . . . . . . . . . . . . . . . . . . . . .
Example with known solution: again Doya . . . . . . . .
Schröder equation . . . . . . . . . . . . . . . . . . . . .
Logistic map

1
2
3
4
5
8

Logistic sequence . . . . . . . . . . . . . . . . . . . . .
Fixed point 𝐿 = 0 . . . . . . . . . . . . . . . . . . . . .
Abelfunction for the Elutin function . . . . . . . . . . .
Halfiterate of logistic operator . . . . . . . . . . . . . .
Another fixed point, 𝐿 = 1 − 1/𝑠 . . . . . . . . . . . . .
Factorial

1
2
3
4
5
9

Physics department . . . . . . . . . . . . . . . . . . . .
Factorial and its fixed points . . . . . . . . . . . . . . .
Regular iteration for factorial . . . . . . . . . . . . . . .
Abelfactorial . . . . . . . . . . . . . . . . . . . . . . . .
Iterates of factorial . . . . . . . . . . . . . . . . . . . .
Exponent to base sqrt(2)

1
2
3
4
5

Superfunction at fixed point 𝐿 = 4 . . . . . . . . . . . .
Abelfunction at fixed point 𝐿 = 4 . . . . . . . . . . . . .
Notations . . . . . . . . . . . .√. . . . . . . . . . . . .
Iterates of exponent to base 𝑏 = 2 . . . . . . . . . . .
Intermediate finish . . . . . . . . . . . . . . . . . . . . .
Exotic iteration and exp to base e1/e

10

1
2
3
4
5
6
11

Exotic iterate . . . . . . . . . . . . . . . . . . . . . . .
Exponent to base exp(1/e) . . . . . . . . . . . . . . . .
Superexponent to base 𝜂 = e1/e = exp2 (−1) . . . . . . . .
Abelexponent to Hernyk base . . . . . . . . . . . . . . .
Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . .
Not only exp . . . . . . . . . . . . . . . . . . . . . . . .
LambertW and zex

1
2
3
4

Holomorphic zex . . . . . . . . . . . . . . . . . . . . . .
SuZex . . . . . . . . . . . . . . . . . . . . . . . . . . .
AuZex . . . . . . . . . . . . . . . . . . . . . . . . . . .
Iterates of zex . . . . . . . . . . . . . . . . . . . . . . .

61

62
64
65
67
71

72
73
78
81
87
89

90
91
94
97
99
103

106
110
113
113
115
117

118
121
123
129
131
133
135

137
140
144
146

12

Sin, super sin and iterates of sin

1
2
3
4
13

Super sin . . . . . . . . . . . . . . . . . . . . . . . . . .
AuSin . . . . . . . . . . . . . . . . . . . . . . . . . . .
Iterated sin . . . . . . . . . . . . . . . . . . . . . . . .
Application . . . . . . . . . . . . . . . . . . . . . . . .
Nemtsov function

1
2
3
4
5
14

1
2
3
4
5
6
7
8
9
10
15

tet

Exponent . . . . . . . . . . . . . . . . . . . . . . . . . .
Heuristic tetration . . . . . . . . . . . . . . . . . . . . .
Properties of tetration . . . . . . . . . . . . . . . . . . .
Cauchy integral . . . . . . . . . . . . . . . . . . . . . .
Taylor expansion at zero . . . . . . . . . . . . . . . . .
Improved approximation . . . . . . . . . . . . . . . . .
Expansion of tet(𝑧) at 𝑧 = 3 i . . . . . . . . . . . . . . .
Asymptotic expansion . . . . . . . . . . . . . . . . . .
Comparison of approximations . . . . . . . . . . . . . .
Implementation . . . . . . . . . . . . . . . . . . . . . .

Natural arctetration

1
2
3
4
1
2
3
4
5
6
7

161
164
168
172
174
175

176
180
184
185
190
193
194
196
199
202
204

Evaluation of arctetration . . . . . . . . . . . . . . . . .
About names . . . . . . . . . . . . . . . . . . . . . . .
Iterates of exponent . . . . . . . . . . . . . . . . . . . .
Lessons of natural tetration and arctetration . . . . . .

√2
Definition . . . . . . . . . . . . . . . . . . . . . . . . .
Again regular iteration √. . . . . . . . . . . . . . . . . .
Arctetration to base 𝑏 = 2 . . √. . . . . . . . . . . . . .
Again iterate exponent to base
√ 2 . . . . . . . . . . . .
Four superexponents to base 2 . . . . . . . . . . . . .
Complex iterates . . . . . . . . . . . . . . . . . . . . .
Not all is done . . . . . . . . . . . . . . . . . . . . . . .

Tetration to base 𝑏 =

16

149
152
157
158
159

ArqNem . . . . . . . . . . . . . . . . . . . . . . . . . .
SuNem . . . . . . . . . . . . . . . . . . . . . . . . . . .
AuNem . . . . . . . . . . . . . . . . . . . . . . . . . . .
Iterates of the Nemtsov function . . . . . . . . . . . . .
End of exotic iteration . . . . . . . . . . . . . . . . . .
Natural tetration

148

205
208
210
213
216

217
217
222
227
231
234
237

Tetration to base 𝑏 > 1

17

1
2
3
18

238

Approximation of tetration near zero . . . . . . . . . . . 238
Various bases of exponent and the iterates . . . . . . . . 242
Dependence of tetration on its base . . . . . . . . . . . 244
Tetration to complex base

1
2
3
19

Fixed points of logarithm . . . . . . . . . . . . . . . . . 247
Tetration to the Sheldon base . . . . . . . . . . . . . . . 249
Shell-Thron region . . . . . . . . . . . . . . . . . . . . . 252
Ackermanns

1
2
3
4
5
20

254

Binary ackermanns . . . . . . . . . . . . . . . . . . . .
Names and notations . . . . . . . . . . . . . . . . . . .
Pentation . . . . . . . . . . . . . . . . . . . . . . . . .
Fixed poins of pentation and future work . . . . . . . .
Comparison of natural ackermanns . . . . . . . . . . . .
Without fixed points

1
2
3
4
5
6
7
21

22

Trappmann function . . . . . . . . . . . . . . . . . . . .
ArcTra, inverse of trappmann . . . . . . . . . . . . . . .
Supertrappmann . . . . . . . . . . . . . . . . . . . . . .
Implementation of SuTra and the testing . . . . . . . .
AuTra, abelfunction of trappmann . . . . . . . . . . . .
Iterates of trappmann . . . . . . . . . . . . . . . . . . .
Relation to other functions . . . . . . . . . . . . . . . .

268
270
276
281
283
287
288
290

Basic results . . . . . . . . . . . . . . . . . . . . . . . .
Future work . . . . . . . . . . . . . . . . . . . . . . . .
Notations . . . . . . . . . . . . . . . . . . . . . . . . .
Afterwards . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgement . . . . . . . . . . . . . . . . . . . . .
Supplement

1
2
3
4
5
6
7

256
257
258
265
266
268

Conclusion

1
2
3
4
5

246

291
293
294
296
297
298

About the cover . . . . . . . . . . . . . . . . . . . . . .
Maple and tea . . . . . . . . . . . . . . . . . . . . . .
Taylor expansion of AuSin . . . . . . . . . . . . . . . .
Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . .
New notations . . . . . . . . . . . . . . . . . . . . . .
Extended abstract . . . . . . . . . . . . . . . . . . . . .
Keywords . . . . . . . . . . . . . . . . . . . . . . . . .

References

298
299
300
301
302
303
304
305

6

Chapter 1
Prologue
I type this Prologue, respecting my teachers ofEnglish, and also those of the Russian Literature. They think, that every scribbling, since
W.Shakespeare, should begin with some Prologue1 , and should finish with some Epilogue. Fig. 1.1: Cloud in Trousers
In addition, the Soviet teachers had promoted the sovietism2 that evena cloud should wear some trousers3 , as it is shown in Fig.1.1.
This prologue appears as a kind of trousers, a wrap, to cover the inside;
where, my results are presented in a naked and provocative form.
This Book is about holomorphic (analytic) solutions 𝐹 of equation

𝐹 (𝑧 +1) = 𝑇 (𝐹 (𝑧))(1.1)

where 𝑇 is some given holomorphic (analytic) function. Also, this Book
is about applications of the solution 𝐹 , its inverse function 𝐺 = 𝐹 −1 andinteresting properties of function 𝑇 𝑛 (𝑧) = 𝐹 (𝑛+𝐺(𝑧)). The given function 𝑇 is called “transfer function”; the solution 𝐹 is its “superfunction”.
The inverse function 𝐺 is called the “Abelfunction”.
The Prologue describes not functions 𝐹 and 𝐺 above, but this Book:
what is it, why is it, how to use it, what is in it, and what is not. I follow
the classical example4 and explain, why this Book is so important.
Readers, who are interested namely in solutions of the equation above
and their physical meaning, may scroll from here directly to Introduction, Chapter 2 at page 20.
1 W. Shakespear. Romeo and Juliet. Prologue. 932/Sh12, Kenkyusba English Classics, 1926.
2 https://mizugadro.mydns.jp/t/index.php/Sovetism

3 https://www.unlikelystories.org/old/archives/cloudintrousers.html

A Cloud in

Trousers by Vladimir Mayakovsky, translated from the Russian by Andrey Kneller. Prologue.
Cited by the state for 2014.09.10.
4 François Rabelais. Gargantua [1] : .. Most noble and illustrious drinkers, and you thrice
precious pockified blades (for to you, and none else, do I dedicate my writings),

7

1

For whom is this Prologue

I type this Prologue not for the Readers. I type this Prologue for those,
who will not read this Book:For Editors, managers, sellers, who deal with thousands books, andwho have need to understand, within a minute, why namely this Book
should be placed at the top in the list of recommended literature.
For the experienced critics, who read only two or three pages, inorder to write the review:What oddity is this: Superfunctions? Whatsort of Superfunctions have we here? And thrust into the world by alaserist! As if we are not satisfied with superconductivity, supersymmetry, superfluidity, supermen and supermarkets! As though threes enoughhad not been cut for paper, and not enough files have been loaded into
internet! As though folks enough of all classes had not tired their fingers
with keyboards! The whim must take a laserist to follow their example!
Really there is such a lot of paper nowadays that it takes time to think
what to wrap in it!

5

For librarians, who need to find for this Book a suitable place, in orderto make it visible at the background of tons of books about superconductivity, supermen, supermarkets and superinflation.
For colleagues, who may wonder, why the laserist, instead of to promote the optical ceramics (for which the big Grants are assigned) deals
with superfunctions (which are yet very far from to get the financial
support), and behaves as a simpleton, who does not know, How to Win
Friends and Influence People.
For relatives and friends, who teach me to live and to promote myresults. They seem to know very well, how to influence people. This bookcould be a kind of medicine against this kind of influenza. However, moredetailed analysis of relations between friends is published separately [23].
For the categories of non-readers mentioned above, I should specify thegenre and topic of this book. The next two sections are dedicated tothis. With such a specification, one may form his/her/its own opinion
about the book, without to read it.
5 N.Gogol. Evenings at a Farmhouse near Dikanka. MARCH 27, 2013. [4] :“What oddity is

this: Evenings on a Farm near Dikanka? What sort of Evenings have we here? And thrust into
the world by a beekeeper! God protect us! As though geese enough had not been plucked for pensand rags turned into paper! As though folks enough of all classes had not covered their fingers
with inkstains! The whim must take a beekeeper to follow their example! Really there is such a
lot of paper nowadays that it takes time to think what to wrap in it.”8

2

Russian version and copyleft

This book had been planned as an
English version of the Russian book,“Суперфункции” 6 . I tried to translate the Russian expressions into English. It happened to be not easy atall; figures 1.1,1.2 prove this.

Fig. 1.2: Problem of translation

I searched the English translations of the classical books cited in the
Russian version. In many cases, the translation is not possible. Often,
it is difficult to find any English equivalents of a reference cited. So, I
give up trying to translate the Book; I re-write it in English.
I reuse some formulas and figures from free sites; in particular, from
Mizugadro and Citizendium; even from Wikipdia. The links are mentioned in the Supplement 4, at page 301. I load images, used in this book,at https://mizugadro.mydns.jp/t/index.php/Category:Book ; itis replica of the previous holder of TORI, https://tori.ils.uec.ac.jp/TORI,
that had been attacked 2013.02.27 (then, the access had been disabled).
It took certain efforts to arrange the clone at https://mizugadro.
mydns.jp/t. The reason of the aggression could be the critical article
prepared for “Physics Today” 7 . Unfortunately, namely this way thepoliticians answer the critics, while they cannot build any logical argumentation. It seems, those politicians pay to administration of ouruniversity some significant amount per each day, while TORI is not available at its original URL. I have no other explanation of the attitude or
our administration: post-factum, they had approved that barbarian attack. I try to make Science, not the criminal investigation, so, I mention
only the scientific point of view on the events. However, I hope, one day,the professional criminalists carefully investigate the case, as well as theorigins of other attacks happened 2013.02.27, 2014.02.27, 2015.02.27,
2016.02.27, 2017.02.27, 2018.03.04 (lated few days since the expected
date 2018.02.27). 8
6 https://mizugadro.mydns.jp/t/index.php/Superfunctions_in_Russian

https//www.ils.uec.ac.jp/~dima/BOOK/202.pdf
https://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic
Publishing, 2014; In Russian.

7 https://mizugadro.mydns.jp/PAPERS/2013physToday.pdf D.Kouznetsov.

Corruption in

Russian science. 2013, preprint

8 https://mizugadro.mydns.jp/t/index.php/Tori_attacked

https://budclub.ru/k/kuznecow_d_j/toriattacked.shtml D.Kouznetsov. TORI attacked, 20139

I allow the free use of my results, and I try to simplify the use. I load theimages, used in the Book, and the codes, use to generate then, at various
sites. I would not specify this in details, but the observations indicate,that the problem is serious. Since year 2014, the strange phenomenon
is observed in the resources of wikimedia: the administrators claim,that publication of my results violate my authors rights. The same,and even worse, happens with results by other authors from the USSR
and, especially, with texts of the Soviet dissidents. The administrators
ignore the permissions, provided by the copyright owners, and remove
the texts and the images, claiming, that they defend the “copyright”.
The similar phenomenon could take place in Nazi Germany. The naziused to arrest, to kill the opponents, and withdrawn their books fromshops and libraries with pretext of defense of the intellectual property.
Fascists killed many authors, and, if the author is already dead, thecensorers claim, that now the State becomes owner of the copyrightproperty. However, the State, after to kill the authors, prohibit the
distribution of their works. Perhaps, a professional historian can correctme, that the German nazi acted in a little bit different way. But everyone
may look at wikisource, and, especially, to its Russian version, and see,that namely this happens at wikimedia since year 2014. Perhaps, theRussian Soviet veterans decided, that the invasion into Ukraine andplundering of Crimea are not sufficient, and begun to plunder also thewiki resources. I mention this in the special statement 9 .
I type the paragraph above in order to stress, that I had issued the permission to distribute my results under the free GNU license; the only
attribution is required. The attempt to “defend” my author’s right in theway described above, is a fraud: several years ago I had allowed the free
distribution, and I provide the links, that confirm this. In particular,
the permission refers to the plots, used in this book and their generators.
I specify the URL of the image at each picture. If some administrator
claims, that use of this image under the “free” license violates my copyright, let me know. I think, such an administrator should be qualified
as liar, impostor, knave, thrive; and his/her/its attempts to defend myauthor’s right in such a way should be qualified as a fraud.
In other words, use the images from this Book, as well as from theRussian version, for free, but do not forget to attribute the source.
9 https://mizugadro.mydns.jp/t/index.php/Kouznetsov,_permissionOpen

letterbyDmitrii Kouznetsov about massive removal of texts and images from Wikimedia projects withpretext of defence to the copyright of the authors. Wed, Mar 18, 2015 at 11:23

10

3

Physical mathematics

While writing this Book, I was asked about its genre. I searched theanswer in various databases, and it seems, the closest qualification is“Physical mathematics” [79, 59, 60].
Term “physical mathematics” should not be confused with “mathematical physics”. The order of words is important. Similar examples exist
in the Quantum Mechanics. Let 𝑃 be momentum of a particle, and 𝑄
be its coordinate; then 𝑃 𝑄−𝑄𝑃 = −i~.
One student could not catch the sense of non-communing operators. She
tried to understand, why operator of creation does not commute with
operator of annihilation. The Professor tried different explanations, thenhe found the strong example for female students; he asked:
Could you explain me, please, why operation TO CLEAN-UP does
not commute with operation TO DIRT-DOWN ?

The student was content, she said:
Thank you, Professor! This is very good example! Now I understandall the Quantum Mechanics!

Professor also got a good lesson; he realised, how to teach Quantum
Mechanics, keeping women and female logic in mind 10 .
Term “mathematical physics” denotes the mathematical exercises withequations that come from Physics.
Term “physical mathematics” means, that the common scientific tools
are used to check mathematical conjectures; the rigorous mathematicaldeduction appears as a possible scientific method, but not an imperative.

4

TORI axioms

In this Book, I use the TORI Axioms. This section retells some resultsfrom publications in “Uspekhi” [68], J. of Modern Physics [84], and “NewInsights into Physical Science”[69]. These results refer to definition of
10 https://pda.anekdot.ru/id/658766 Bogorad Victor. Мысли, Мысли. (2013, In Russian)https://mizugadro.mydns.jp/t/index.php/Female_logic D.Beklemishev.

Female logic.

(2013)

https://www.ams.org/notices/201309/rnoti-p1156.pdf J.M.Deshler, E.A.Burroughs. Teaching Mathematics with Women in Mind. Notices of the AMS, V.60, No.9, p.1156-1163.

11

term “Science”. The TORI Axioms appear as parts of the definition ofthis term.
Science is kind of knowledge, activity and research, based on
concepts, that have all the six properties below:
1. Applicability: Each concept has the limited range of validity, dis-tinguishable from the empty set.
2. Verifiability: In the terms of the already accepted concepts, somespecific experiment with some specific result, that confirms the concept,
can be described.
3. Refutability: In the terms of the concept, some specific experimentwith some specific result, that negates the concept, can be described.
4.

Self-consistency:

No internal contradictions of the concept are

known.
It the range of validity of a newconcept intersects the range of validity of another already accepted concept, then, the new concept either reproduces the results of the oldconcept, or indicates the way to refute it. (For example, the estimate of
the range of validity of the old concept may be wrong.)

5. Principle of correspondence:

Mutually-contradictive concepts may coexist; if two
concepts, satisfying requirements 1-5 above have some common range ofvalidity, then, in this range, the simplest of them has priority.
6.

Pluralism:

The 6 properties, requirements above are TORI axioms. They are
specified also at https://mizugadro.mydns.jp/t.
The main results, that had been presented at TORI and related to thisBook, are published in Scientific Journals [84, 86, 89, 92]. In order
to simplify the referencing, the axioms above needed a special name.
Having poor fantasy, I just keep the initial abbreviation, TORI, thatmeans Tools for Outstanding Research and Investigation. The Russiantransliteration “ТОРИ” has similar meaning (“Теоретические Основы
Революционных Исследований”). Many results, reported at the conferences and in the scientific journal, look so similar, that it is difficult
to find, what namely new and interesting does each of them suggest.
12

They barely satisfy the TORI axioms, if at all. The non-trivial resultsappear to be “outstanding”, “revolutionary”, in the sense, that they stay
a little bit out of the main trend of scientific research.
Articles of site https://mizugadro.mydins.jp/t collect tools: concepts, formulas, algorithms, pictures, that I consider to be “outstanding” and useful in the scientific research. I assume, that the “instanding”
researches are already well represented in Citizendium, Wikipedia andscientific journals, and there is no need to repeat them again. In particular, I load to TORI generators of figures, used in this book. The idea is,that each code can be downloaded by the Reader, compiled, executed,
and, if necessary, modified for the new (and, perhaps, unexpected forme) applications.
I load the generators also to other sites,in order not to make an impression, thatthe politicians may completely block theaccess to my results attacking anotherserver. In this sense, Manuscripts don’tburn [8].
In the Supplement, I suggest the links toother sites for the case if some server isFig. 1.3: Server felt down
not reachable. The same, for the case, if
it suddenly falls down, as it is shown in Figure 1.3.
With links, supplied to figures, the Reader can use and develop theformalism of superfunctions just from the place, until I have advanced.
Just pick-up the figure you like and load its generator.

5

What for this book is

Since publication in 2009 of the article about holomorphic tetration [54],I expected, the mathematicians can do the rest of the job by themselves.
That publication refers to the special case of equation (1.1), with 𝑇 =exp; and the natural tetration as solution is considered, 𝐹 = tet.
Many years passed by since [54], but superfunctions did not appear indescriptions of the algorithmic languages. Neither tetration, nor superfactorial [66], nor other superfunctions I had reported appear as built-in
13

functions. Until now, the function Nest [99] in language “Mathematica”
requires, that the number of iterate is integer. If one approximates somebell-like function, then, usually, the fantasy of colleagues does not go
farther than the Gaussian exponential, hyperbolic secant or Lorentian.
One could use also, for example, the half-iterate of the exponential [10],id est, such function 𝑓 , that 𝑓 (𝑓 (𝑧)) = exp(𝑧), instead of the exponent
in 2/(e𝑥+𝑒−𝑥 ), as well as other non-integer iterates of various functions,but one needs the formalism of superfunctions for the evaluation. This
Book describes such a formalism.
Most of the Mathematical Analysis in schools and universities is built-up
on the first three ackermanns: addition, multiplication, exponentiationand their inverse functions in various combinations. This arsenal could
be greatly extended, including, for example, the 4th ackermann, which is
tetration, or the 5th ackermann, that is pentation. However, one needs
formalism of superfunctions to evaluate tetration and pentaiton [64, 90]
When physicists analyse the nonlinear response of a medium, they try
to make the sample optically-thin; otherwise, the re-absorption and saturation lead to mistakes [42, 43, 44, 45, 46, 47]. For the thin sample,
the precision of measurements is low. Superfunctions allow to workwith thick samples, and still reconstruct behavior of intensity inside thesample [85, 86].
Many researchers believe, that namely their work is especially important. I am not an exception. From my point of view, tetration andarctetration, as well as other superfunctions and abelfunctions, will be
important for science since century 22; so important, as the exponential
and logarithm are important since century 19. This Book explains, how
to calculate superfunctions and what can one do with them.
When the fundamentals of superfunctions had been formulated, one of
my coauthors, Akira Shirakava [38] had declared, that it is not possible, to recover distribution of intensity of light in the amplifier fromits transfer function. [75]. Then, the algorithm for this recovery had
been presented [61, 85, 86], and Shirakava said, that the formalism of
superfunctions is too complicated [76]. Through all the mutual misunderstandings, I consider Shirakawa as great specialist in fibre lasers, andI take into account his complains. I try to make this book popular.
So, I cite not only scientific articles. Also, I try to avoid complicatedmathematics, at least in some figures. I still suppose, that the Reader
14can distinguish an Integral from a Logarithm, and has some idea, whatis square root of minus unity. If the Reader, in addition, already had
seen any Cauchy integrals, this is a good advantage.
In order to simplify the reading, I try to avoid long sentences and digressions, although not always it is possible. I suggest the example:
Happy is the traveller who, after a long and heavy journey, with its coldand humidity of air-conditioners, with long waiting-lines at the passport
control, curious to under ware the special control officers, delayed flights,
lost luggage, going to the opposite direction taxi drivers, sees at last thefamiliar roof with its lights approaching to meet him. And there rise
before his mind the familiar rooms, the delighted outcry of the servantsrunning out to meet him, the noise and racing footsteps of his children,and the soothing gentle words interspersed with passionate kisses thatare able to efface everything gloomy from the memory. Happy the manwith a family and nook of his own, but woe to the bachelor!
Happy is the researcher who, passing by the strange results, paradoxes
and their wrong, inconsistent interpretations, attaches himself to phenomena that display the loftiest virtues of scientific achievements, who,from the great whirlpool of figures flitting by him daily, has selected
only the few exceptions, for which the answers are already known; who
has never tuned his research to a less exalted key, has never stooped
from his pinnacle to really new and unexpected phenomena. His fairportion is doubly worthy of envy; he lives in the midst of them as in the
midst of his own family and, at the same time, his fame resounds far
and wide. He clouds men’s vision with enchanting incense; he flattersthem marvellously, covering up the gloomy side of science and life, andexhibiting to them the noble man. All run after him, clapping theirhands and eagerly following his triumphal chariot.
They call him a great world-famed scientist, soaring high above every
other genius as the eagle soars above the other birds of the air. Youngardent hearts are thrilled at his very name; responsive tears gleam inevery eye. . . . No one is his equal in power — he is a God! But quite
other is the portion, and very different is the destiny of the writer whosees and reveal the phenomena strange, that are out of the commonly accepted theories and contradict to the obvious commonly-accepted common sense, calling for the revision or at least some critical analysis of thewidely recognized results, that already have assigned the highly prestigious awards and huge grants and foundations.. En fin, he’ll not escapefrom the scientific council, who keep the old paradigms and any doubt
consider as a sin, as a kind of heresy.. Without compassion, such a researcher is left by the roadside like the traveller without a family. Hardis his lot and bitterly he feels his loneliness. [5]I promise, that the above is longest digression in this Book. Trying tokeep the Book thin, the philosophic exercises about the place of Sciencein the Human knowledge [68, 69] are published separately. It is not15possible to cover Everything (id est, the Full Set). Since the beginning,
I try to specify, what is in the book, and what is absent in the Book.

6

What is present in the Book

This Book suggests the general approach for construction of superfunctions and various algorithms of their evaluation. Superfunction is solution 𝐹 of equation (1.1); I repeat it here: 𝐹 (𝑧 +1) = 𝑇 (𝐹 (𝑧)). Thisequation is called “Transfer equation”, and the given function 𝑇 is called“transfer function”.
In this book, I describe various algorithms for evaluation of superfunctions. The choice of the algorithm depends on the fixed points of thetransfer function, which are solutions 𝐿 of equation 𝑇 (𝐿) = 𝐿; and,also, depends on the additional requirements: what properties do we
expect from the required solution 𝐹 of equation (1.1). These properties, being postulated, should provide the uniqueness of the solution,
but still should allow its existence and should indicate some way(s) of
the evaluation.
The inverse of superfunction 𝐹 is called here “abelfunction” 𝐺 = 𝐹 −1 .
The abelfunction satisfies the Abel equation 𝐺(𝑇 (𝑧)) = 𝐺(𝑧)+1. In thisbook, I consider many examples of the transfer function 𝑇 , superfunction 𝐹 and the abelfunction 𝐺. I suggest ways of the efficient evaluation.
Efficient means, that the functions (and corresponding figures) are computed in the real time and with good precision. With use of the complex
double arithmetics, the superfunctions and abelfunctions are evaluatedwith 14 decimal digits. I load the figures together with their generators
to https://mizugadro.mydns.jp/t/index.php/Category:Book; thereaders can check them with C++ and Latex.
By default, I assume, that the parameters have real values, while thearguments of the functions may have complex values. For the illustrations, the complex maps are essential. In particular, the complex mapof the natural tetration is used at the front cover of this Book.
I am more physicist, than mathematician. For physical applications,the real values of the argument are of special interest. Usually, thereal values allow the straightforward interpretation and, sometimes, theapplication. For this reason, I supply also the graphics for the real valuesof the argument.

16

7

What is absent in the Book

In this book, there are no accurate specifications, which values are allowed for parameters and arguments of the functions considered. I specify them only in the cases, when they are essential for the consideration.
For each expression, the readers are supposed to examine the range ofapplicability as the exercise. The reader is not a vessel to be filled, but
a candle to be lighted 11 .
I try to define functions in such a way, that at least some vicinity ofthe positive part of the real axis belongs to the range of holomorphism;
although, not always this is possible. This book seems to be the firstmonograph with description of efficient methods of evaluation of superfunctions and abelfunctions, and this should be considered as an excuse
for the heuristic style of presentation of the results.
For the same reason, in this Book, there are no rigorous proofs of convergence, existence and uniqueness of solutions. In particular, the deductions, elaborated in collaboration with Henryk Trappmann, are notincluded here. The Readers who like the deep drilling 12 are invited todownload our original articles [61, 67, 74, 80, 87], plumb through the
deduction, expanded with the Saxon style of pedantic Henryk, drill theexamples and ask questions. Indications of the errors are of especial
interest.
From my point of view, the theory of superfunctions is new branch of theMathematical Analysis, and the serious drilling and plumbing require
efforts of some institute with several research groups in order to bringit to the beautiful and rigorous shape, like the theory of the differential
and Integral calculus and other mathematical analysis. Here, I deal withfunctions of single variable; the generalisation to multidimentional caseis not presented.
History to development of theory of superfunctions and abelfunctions
begins approximately since century 19, and can be also interesting, but
11 https://quoteinvestigator.com/2013/03/28/mind-fire/ The Mind Is Not a Vessel That
Needs Filling, But Wood That Needs Igniting. March 28, 2013. .. None of the examples camewith citations:
Education is the kindling of a flame, not the filling of a vessel —Socrates
Education is not the filling of a pail, but the lighting of a fire. —William Butler Yeats
Education is not the filling of a pail, but the lighting of a fire. —Plutarch
The mind is not a vessel that needs filling, but wood that needs igniting. —Plutarch ..
12 Importance of the deep drilling can be illustrated with the limerick below:
One Chinese colleague called Lee
Was drilling his girl at the sea;
She told him: Stop plumbing,Somebody is coming!

And Lee had replied:That’s me!

17

it falls out of the topic of this Book. The overviews of the pre-historic
publications can be found in the Internet with the following keywords:
Abel function, iterated functions, tetration, superexponential,
superfunction. Some of them can be considered as scientific, accord-

ing to the definition above; I mean, the TORI axioms. This Book is
dedicated to the Superfunctions, not to TORI axioms, so, these axioms
are presented here only schematically; however, the more detailed description is available [68, 69, 102, 103, 104].
In this Book, there is no detailed discussion (and sometimes nor evencitation) of publications of colleagues, who tried to evaluate the superfunctions during the last couple of centuries. In order to see, what had
been done in the past centuries, one can look at the publications by colleagues: Neils Henryk Abel [3], Peter Fatou [6], Hellmuth Kneser [10],Jean Ecalle [17], G.Szekeres [11], Peter Walker [24]; more references aresuggested in year 2014 by J.Bonet and P.Domanski [91]. In addition,some links are listed in the Supplement (page 301).
I see analogy with manuals
on Geography and Astronomy, that have no need to
begin with the historic excursus, describing the Flat
Earth (Figure 1.4), nor
the Geocentric model byPtolemy.
Perhaps, some
Readers may consider suchan analogy to be too pretentious, too ambitious; but
the methods, reported be- https://commons.wikimedia.org/wiki/File:Flammarion-color.png
fore century, 21 are, indeed, https://en.wikipedia.org/wiki/Flat_Earthnon-efficient; so inefficient, Fig. 1.4: Earth in the old model of the Universethat nobody could plot acomplex map of any non-trivial superfunction with methods reported
in century 20. Henryk Trappmann attributes this to my routine https://mizugadro.mydns.jp/t/index.php/Conto.cin, which is now available also at various wikis. I think, the matter is not my plotter, but
the beautiful and efficient representations of functions, that allow thequick and precise evaluation. In principle, these algorithms can be programmed at Mathematica, even at Maple, and still generate the same
figures; although the efficiency of Maple (and, especially, Maple-10) ingenerating of figures allows to use that software as an illustration to the18known proverb “to press a key, to have a tea”, as it is described in the
Supplement, page 299. I do not consider that poem as a part of thisBook, but the poem and figure 22.1 explains, why the most of figuresin this Book are generated as the direct EPS graphics from the C++
programs.
I do not describe here algorithms, suggested after year 2015; for example,
those by William Paulsen and Samuel Cowgill [93, 94, 95, 96, 97].
I do not include into this Book the codes of the programs (figure 22.1
is an exception), used for evaluation of functions and plotting graphics.
The algorithms are simple; any student, after a semester of programming in any language, that supports the complex arithmetics, can easy
program the same, following the description from this Book. At figures,
I indicate the URLs, where the code is loaded.
Some results of this book can be reproduced with the Schröder equationand the Schröder functions. In this Book, there is no detailed description of properties of the Schröder equation, nor analysis of the Schröderfunctions; these functions are only mentioned. I could not construct
any example, that can be solved with Schröder functions, but cannot be
solved with superfunctions. Some properties of the Schröder equation
can be found athttps://mathworld.wolfram.com/SchroedersEquation.html andhttps://en.wikipedia.org/wiki/Schroeder_equation

Preparing the results for the Book, I tried to refute each of the conjectures suggested. In principle, it is possible. It is sufficient to construct
any example of a transfer function 𝑇 such that the superfunction 𝐹 ,constructed by the methods, described in the Book, does not satisfy thetransfer equation (1.1). Or to construct two different superfunctions,that satisfy the additional conditions, that are declared to provide theuniqueness. My attempts to reject my claims failed. I could not find in
the literature any rejecting example, nor I could construct it by myself.
Only for this reason, the rejecting, refuting examples are not includedin the Book.
Many colleagues, as myself (since the childhood), prefer to watch pictures in a book, rather than to read it. Therefore, I try to illustrate eachpage with at least one figure. While, this goal is not yet achieved, but I
hope to approach it closer in the future editions.

19

Chapter 2

𝐹 (0)Introduction

𝑇 (𝐹 (0)) = 𝐹 (1)Iterates and Superfunctions arise naturally
Fig. 2.1: Snowball of mass 𝐹 (0)at description of a sequence of similar transgets mass 𝑇 (𝐹 (0)) = 𝐹 (1) after
formations of any quantity in a homoge- to roll once from the hill.
neous physical system. This transform mayrefer to accumulation of mass by a snowball, that rolls down the hill,covered with sticky snow (Fig. 2.1). This may be attenuation of a shock
wave in one of the successive sections of automobile muffler. The transform may refer to change of intensity of light, passing one section of alaser amplifier or a saturable absorber. I hope, the Reader can suggest
more examples of this kind.
Let the state of the system before the transformation be described withparameter 𝑥, and, after transformation - with parameter 𝑦 ; and let thesetwo quantities be related with 𝑦 = 𝑇 (𝑥). Then, function 𝑇 is called“transfer function”. It describes the transformation of the signal in thesystem.
System, that performs the transformation, can be called “filter”. Transformation of signal 𝑥 in a filter with transfer function 𝑇 can illustrated
with expression

𝑥 → filter → 𝑦 = 𝑇 (𝑥)(2.1)

Output of one filter can be directed to the input of another identicalfilter. This can be expressed with𝑥 → filter → 𝑇 (𝑥) → filter → 𝑇 (𝑇 (𝑥))(2.2)

One can write the similar expressions for combination of 3 filters and soon. In these notations, the number of filters combined is supposed to
be a positive integer. The main idea of this Book is, that the numberof filters has no need to be integer. In this chapter, the notations aresuggested to describe this.
20

1

Use of superscript

From the point of view of physics, every iterate describes some combination of identical filters. Let each filter be characterised with the sametransfer function 𝑇 . If we have signal 𝑧 at the input, then, at the output,
the signal is 𝑇 (𝑧).
As it is mentioned in the Preamble of this chapter, the output of a filter
can be directed to another similar filter, and the transfer function of theresulting combination can be expressed with

𝑇 (𝑇 (𝑧)) = 𝑇 2 (𝑧)(2.3)

In the right hand side of equation (2.3), the notation with superscript
is used. I am not first to suggest this notation; Walter Bergweiler had
used such a notation in the past century [26]. If there is superscript at
name of some function, and the expression in this superscript can beinterpreted as a number, then it is number of the iterate. The zeroth
iterate of a function is supposed to be the identical function (its value
is the same as value of its argument):

𝑇 0 (𝑧) = 𝑧(2.4)

First iterate of some function 𝑇 is the same function 𝑇 , and the minus first iterate is the inverse function. For example, at the sinusoidaltransfer function 𝑇 = sin, we havesin−1 (𝑧) = arcsin(𝑧)(2.5)

sin0 (𝑧) = 𝑧(2.6)

sin1 (𝑧) = sin(𝑧)(2.7)

sin2 (𝑧) = sin(sin(𝑧))(2.8)

sin3 (𝑧) = sin(sin(sin(𝑧)))(2.9)

and so on. In some textbooks, the notation is used (but not declared),when expression sin𝑎 (𝑧) has sense of sin(𝑧)𝑎 . Such a notation leads

1; but,
sin(𝑧)
according to another commonly used notations for the inverse function,
it should mean sin−1 (𝑧) = arcsin(𝑧).
to confusions: at 𝑎 = −1, expression sin𝑎 (𝑧) might mean

In such a way, in this Book, the expression in the upper superscript at
the name of a function is interpreted as number of iterate. However,if the superscript is just “prime”, almost vertical stick, it indicates thederivative; for example, sin′ = cos; it is also usual notation.
21

filter

𝐹 (0)𝑇 2 (𝑓0 )𝑇 (𝑓0 )𝑓0

𝐹 (𝑧)filter𝐹 (1)𝑇 3 (𝑓0 )
filter𝐹 (𝑧 +1)𝐹 (2)... −→𝐹 (3)𝑧Fig. 2.2: Combination of filters, each of them has transfer function 𝑇

2

Transfer equation

The use of the superscript allows to illustrate expressions (2.1) and (2.2)
with figure 2.2. The figure shows the combination of identical filterswith transfer function 𝑇 each. I assume 𝑇 to be some given, known
holomorphic function. The range of holomorphism is supposed to be
wide enough, to cover the needs of the following consideration.
In the left hand side of figure 2.2, at the entry of the first filter, let thesignal be characterised with some fixed value 𝑓0 . Then, after to pass
the filter, the signal becomes 𝑇 (𝑓0 ); after to pass the second filter, it
becomes 𝑇 2 (𝑓0 ) = 𝑇 (𝑇 (𝑓0 )), and so on. These quantities are specified
above the vertical bars, that mark the end on one filter and beginningof the next filter.
The notation can be even shortened, to count number of passes of thefilter. Let 𝑓0 be 𝐹 (0); let 𝑇 (𝑓0 ) be 𝐹 (1); let 𝑇 2 (𝑓0 ) be 𝐹 (2), and soon. In such a way, function 𝐹 of non-negative integer argument can be
defined.
Now I need the strong assumption:
Let the filters, inside, be uniform, and act as some kind of continuous
homogeneous nonlinear medium (for example, amplifier), that transfers
the signal by some fixed (although, may be, not yet known) way; but,after to pass each section of the combined filter, the signal is transformed
with known the same known transfer function 𝑇 .
We may consider coordinate 𝑧 along this combined filter, and treat it asa continuous medium. It is convenient to choose the length of a single
filter as a unit to measure the coordinate along the combined amplifier.
Then, 𝐹 (𝑧) may have sense of signal at coordinate 𝑧 ; for integer valuesof 𝑧 , values of function 𝐹 (𝑧) are already known. The question is:

How to define, determine, evaluate 𝐹 (𝑧) for non-integer 𝑧 ?

22Assume, for some non-integer position 𝑧 , for example, between zero andunity, the signal is 𝐹 (𝑧). Then, in the uniform medium, after a passlength unity along the composed filter, the signal should be transformed
with the same transfer function 𝑇 . This can be expressed with the transfer equation (1.1) mentioned in the Prologue. I repeat this equation:

𝐹 (𝑧 +1) = 𝑇 (𝐹 (𝑧))(2.10)The transfer equation (2.10) specifies, how does the value of function 𝐹changes, while its argument gets the unity increment. In such a way,
figure 2.2 indicates the physical meaning of the main equation in thisBook; the figure suggests the physical implementation of the transfer
equation (2.10).

3

Values of the argument

In this Book, I assume, that the signal, transferred through the filter,is characterised by a single number, real or complex. In principle, term“transfer function” allows generalisation to the multidimensional case;then, the transfer function appears as a functional, and its argument
may have sense of a vector or a function. As it is mentioned in the Preface, here I consider only the case of a single variable: it happened, thateven the single-dimensional case causes a lot of confusions at the internet forums and discussions. It is methodically-incorrect, to develop themultidimensional generalisations, while the colleagues have doubts evenin the single-dimensional case. According to the definition of science[68, 102] in the Prologue, first, the results about the single-dimensionalcase should be presented, that still refute the commonly accepted pointof view, that the recovery of the signal inside a homogeneous systemfrom its transfer function is not possible [72, 73, 75, 76].
One of objections refers to the multitude of solutions 𝐹 of equation(2.10). Consideration of complex values of argument of the transfer
function allows to reduce the class of possible solutions (and sometimes
even provides the uniqueness of the solution). In such a way, consideration of complex argument is more important, than analysis of somemultidimensional signal. For this reason, in this Book, I assume, that theargument of the transfer function is a complex number. Also, I assume,that the transfer function 𝑇 is holomorphic, at least at some vicinity ofsome part of the real axis. In the most of cases, 𝑇 real-holomorpic andgrowing function (at least at some part of the real axis).
23

4

Nest

For iterates of functions, in algorithmic language Mathematica, there is special routine Nest. It requires 3 arguments. The first one indicates the name of function thatshould be iterated. The second argument indicates the Fig. 2.3: Nest
initial value at the iterates. The third (and last) argument specifies the number of iterate. The call looks as follows:

Nest[𝑓, 𝑥, 𝑧](2.11)In Mathematica, the arguments of a function appear in squared parenthesis; and expression (2.11) means 𝑓 𝑧 (𝑥). (However, term “Nest” mayhave also different meaning, as it is shown in figure 2.3.)
At least until year 2017, the implementation of Nest requires, that thelast argument can be simplified to a positive integer constant. Even 0th
and minus first iterates are prohibited. Perhaps, the designer assumed,that only the integer number of iterate may have sense and meaning.
Actually, the non-integer iterates do have sense, as it is mentioned in
the preamble of this chapter. One more example is considered in thenext section.

5

Fibre amplifierThis section describes the example, that shows the sense of the noninteger iterates. This example refers to the fibre amplifier. I type “fibre”
in order to indicate, that the signal is confined in two directions, andonly the power in the fibre as function of the coordinate along it issubject of consideration. A lot of physical effects are dropped out inthis consideration: change of the spectral content of the signal, selfpeaking, spontaneous emission, etc.. So, I consider here the simplest
case, but for this model I want to get the exact solution.
Assume, some Manufacturer gives to some Physicist a piece of one meterlong of the optical fibre amplifier, together with the system of pumping,
and asks the Physicist to investigate, how the power of light inside grows
during the amplification, but Manufacturer does not allow Physicist to
cut the fibre to measure, what is inside. Physicist knows the only, thatthe fibre is uniformly pumped. We may assume, that some system oflateral delivery of pump is used [32, 33, 34, 41].
Physicist can measure the transfer function 𝑇 of this fibre. With thistransfer function, Physicist can say, what should be the transfer functionof similar piece of fibre of length 2 meter. Assuming, that the source24

of pump is delivered together with each piece of the fibre, the transfer
function of the piece of two meter is 𝑇 2 ; that of the 3 meter piece should
be 𝑇 3 and so on; if 𝑧 pieces of the fibre are combined, then, the transfer
function is 𝑇 𝑧 .
Iterates 𝑇 𝑧 are pretty clear, while 𝑧 is integer. But how about noninteger values? For example, what is transfer function to the piece ofhalf meter long?
As it is mentioned in the previous section, in Mathematica, the transfer
function of piece of 𝑧 meter length could be expressed with 𝑇 𝑧 (𝑥) =Nest[𝑇, 𝑥, 𝑧]. Unfortunately, such an expression is not yet interpreted
correctly at non-integer 𝑧 .
How to express the non-integer iteration of a given transfer function?
These questions can be analyzed with the transfer equation (2.10). The
preliminary analysis is suggested in the next section.

6

Transfer equation and the Abel equation

Through this book, the transfer equation (1.1),(2.10) is repeated (andused) again and again, and, in particular, here:(︀)︀𝐹 (𝑧 +1) = 𝑇 𝐹 (𝑧)(2.12)

I remind, that 𝑇 denotes the transfer function, 𝑧 may have sense ofcoordinate along direction of propagation of some signal, and function
𝐹 expresses dependence of the signal on this coordinate. (However,coordinate 𝑧 may have also any other meaning.)
In Laser Science, term “signal” denotes the wave (light), that is amplified; even if no information is transferred with this light. In this book,the length is measured in units of the length of the amplifier. The generalisation for the arbitrary units is straight-forward.
For given transfer function 𝑇 , solution 𝐹 of the transfer equation (2.12)
is called “superfunction”. The inverse function 𝐺 = 𝐹 −1 is called “Abelfunction” or “abelfunction” for the same transfer function 𝑇 . The abelfunction 𝐺 satisfies the Abel equation

𝐺(𝑇 (𝑧)) = 𝐺(𝑧) + 1(2.13)

Equations (2.12) and (2.13) can be deduced from each other other by
the change of variable. Just replace 𝑧 to 𝐹 (𝑧) in equation (2.13) andapply function 𝐹 to both side of the resulting equation. The Readers
are united to make this exercise or to look for it at some Wikipedia [98].
(I hope, the Soviet veterans, that remove articles about the USSR, will
be stopped before they begin to vandalise the Abel equation.)25

Assume, that the superfunction 𝐹 and the abelfunction 𝐺 are somehow
constructed. This gives key to the iterate 𝑇 𝑧 of the transfer function,and the number 𝑧 of the iterate has no need to be integer. This iterate
can be expressed as follows:

𝑇 𝑧 (𝑥) = 𝐹 (𝑧 + 𝐺(𝑥))(2.14)In order to show an example of application of formula (2.14), return tothe story about Manufacturer and Physicist from the previous section.
Assume, the Physicist has found the physically-meaningful solution 𝐹of the transfer equation (2.12), and has constructed the inverse function,
id est, abelfunction 𝐺 = 𝐹 −1 . Then, Physicist can express the transfer
function 𝐹 of the amplifier of arbitrary length 𝑧 by formula (2.14).
For transfer function 𝑇 , its superfunction 𝐹 and the abelfunction 𝐺
appear as two sponges of a wrench for some screw, as a tool, that allows
to rotate the screw for any rational angle.

7

Multiplicity of solutions

Solution 𝐹 of equation (2.12) is not unique. One can reduce multitude
to solution, specifying its value at zero (or at any other point the Reader
likes); id est, choose some number 𝐹0 and add the requirement that

𝐹 (0) = 𝐹0

(2.15)

In many cases, the constant 𝐹0 does not affect the shape of iterates of
the transfer function; transform˜𝐹˜ (𝑧) = 𝐹 (𝑧 +𝑥0 ) , 𝐺(𝑧)
= 𝐺(𝑧)−𝑥0(2.16)
˜ into equation (2.14) gives
for some 𝑥0 and substitution 𝐹 → 𝐹˜ , 𝐺 → 𝐺
the same iterate, as the initial 𝐹 , 𝐺 do.
However, even after addition of requirement (2.15), the solution is notunique. If 𝐹 is solution, superfunction, then another solution (anothersuperfunction) can be constructed with(︀)︀𝑓˜(𝑧) = 𝐹 𝑧 + 𝜃(𝑧)(2.17)

where 𝜃 is periodic holomorphic function with period unity. At 𝜃(0) = 0,even condition (2.15) is preserved.
The different superfunctions 𝐹 , with their abelfunctions 𝐺 = 𝐹 −1 , givedifferent iterates of the transfer function 𝑇 . For this reason, one may
think, that the non-integer iterates of a function have no meaning. In
particular, in century 20, the colleagues
had believed, that the half iter√
ation of factorial, denoted with ! = Factorial1/2 , has no meaning.
26

√Function! is especially interesting, because
since 1950, it is used as logo of the Physics Department of the Moscow State University [36] 1 .
That logo is shown in figure 2.4, borrowed from
Russian publication [37]. Only in 2010, the physical and mathematical meaning of this iterate and√meaning of this logo had been reported [62], with Fig. 2.4: ! as logo
formalism of superfunctions.
The uniqueness of the superfunction of factorial (and, therefore, thatof the half iterate of factorial) is provided by the additional requirement of holomorphism and behaviour at infinity: transformation (2.17)
reduces the range of holomorphism of superfunction and that of thereconstructed non-initeger iterates. Complex numbers are essential.
The problem of evaluation of non-integer iterate in some physicallymeaningful way arises not only at the phenomenological considerationof an idealised amplifier. Similar problem appears at the analysis ofstability of jets; in the simple (single-dimensional) approximation, theappearance and disappearance of instability by the Pomeau-Mannevillescenario [19, 53] can be described with some specific quadratic transfer
function. Similar equation arise at the analysis of stability of attractors[20]. At least for the single-dimensional models, one can construct abelfunction and superfunction [70], and, hence, the non-integer iterates of
the transfer function.
Observation of similarities in construction of superfunctions allows toformulate the problems about superfunctions and the goal of this Book.
This is suggested in the next section.
1 According to the results of the student vote, winners were the icon with the image of the lever,raising the Earth, and the current from the well-known emblem of the root of factorial, inscribedin the letter Φ. This icon is created by the student at Department of Biophysics, A.Sarvazyan; it
is attracted by its simplicity and expressiveness. Then the icon emblem has undergone harsh criticism from the Faculty of management, since it has no physical meaning, mathematically absurd
and ideologically vacuous. Yielding to the requests of the Komsomol, the party committee (Professor A.I.Kuzovnikov) agreed that the icon to release a series of small, only one year, so that in the
future to create a real symbol for the Faculty students. Now, paradoxically, the emblem of the firsticon was included in the official form and faculty printing. ..

The Russian original [36]:По итогам студенческого голосования победителями оказались значок с изображением
рычага, поднимающего Землю, и нынешний с хорошо известной эмблемой в виде корняиз факториала, вписанными в букву Ф. Этот значок, созданный студентом кафедры
биофизики

А.Сарвазяном,привлекал

своейпростотой

ивыразительностью.

Тогдаэмблема этого значка подверглась жесткой критике со стороны руководства факультета,поскольку она не имеет физического смысла, математически абсурдна и идеологически
бессодержательна. Уступая просьбам комсомольцев, партком (профессор А.И.Кузовников)согласился на то, чтобы значок выпустить малой серией, только на один год, с тем, чтобыв дальнейшем создать настоящий символ для студентов факультета.

Сейчас, как непарадоксально, эмблема первого значка вошла в официальный бланк и печать факультета.

27

8

Formulation of the problem

Suppose we have some transfer function 𝑇 , holomorphic in a wide rangeof values of the argument, and we are interested in its iterates.
What additional conditions should be imposed on superfunctions 𝐹 (orto the abelfunction 𝐺 = 𝐹 −1 ) as solution of the transfer equation (2.12)(or that of the Abel equation (2.13)) to provide the uniqueness of thesuperfuncton?
How to evaluate the superfunctions and the abefunction, that satisfying
the conditions chosen?
How to verify that namely this solution has the physical meaning?
The goal of this book is to answer these questions. I try to represent theanswers in the most simple and explicit form. In this book, I retell thepreviously published articles [54, 61, 63, 70, 66, 80, 92, 89, 86]; the idea
is to release the Reader from the need to drill the original publications.
The next chapter suggests examples of superfunctions and corresponding
abelfunctions; other chapters suggest general algorithms for calculation
superfunctions and the application to physical problems. Description of
the state of a physical system with a single parameter is already a strong
approximation, and at least in this case, it is desirable to represent thesolution in a simple and exact form.
Perhaps, I should discuss meanings of term
“simple”. Applying to a function, this term
may get new meaning, indicating, that values of the function belong to some finite set
2. In the similar way, the adjective “dry” significantly changes its meaning, being applied
to a vine or Martini, as it is shown in Fig.
2.5. In this Book, term “simple” indicates,that the function is easy to understand, its
definition is short and its implementation isfast. As for the set of values, the functions
are supposed to be holomorphic, and, therefore, should have values from the continuous
set of complex numbers.

Devchonka buys
Dry Martini
Fig. 2.5: Dry martiniOn this point I finish the general speculations about superfunctions andturn to the specific examples. In the following chapter, few transfer
functions, their superfunctions and abelfunctions are considered.
2 https://en.wikipedia.org/wiki/Simple_function In the mathematical field of real anal-ysis, a simple function is a real-valued function over a subset of the real line, similar to a step
function.
28
1Chapter 3
Table of superfunctions
Before to calculate the superfunctions and abelfunctions for transfer
functions of general kind, it worth to see the cases, when the superfinction and the abelfunction can be easy expressed in terms of alreadyknown special functions. These examples are mentioned in this chapter.
It can be considered as continuation of the Inroduction: the most offunctions mentioned here are known since the school course of algebra.
Superfunction 𝐹 for the transfer function 𝑇 is solution to the transfer
equation (2.12); I repeat it here:

𝐹 (𝑧 + 1) = 𝑇 (𝐹 (𝑧))As the main idea of this Book are described above, the table of superfunctions 3.1 already has sense. This table is important and deserves
the special (although very short) chapter.
In the next section, several known solutions are presented in Table 3.1.
Then, in the following sections and chapters, these functions are considered with more details.

1

Old trick

There is analogy between the table of superfunctions and tables of integrals. In both cases, the direct operation is, in certain sense, moredifficult, than the inverse operation. If one knows the indefinite integralof some function, and this integral is expressed in terms of elementaryfunctions in a compact form (the writing fits the width of the columnof the table), then, one can calculate the integrand with known rulesof differentiation. In the similar way, if the superfunction 𝐹 and theabelfunction 𝐺 = 𝐹 −1 are known, the transfer function can be expressedwith

𝑇 (𝑧) = 𝐹 (1 + 𝐺(𝑧))29Table 3.1: Examples of superfunctions 𝑓 :(︁)︁
𝑇 (𝑧) = 𝑓 1 + 𝑔(𝑧)𝑔(𝑧) = 𝑓 −1 (𝑧)

𝑇 (𝑧)

𝑓 (𝑧)

1

𝑐

𝑐

2

𝑧 +1

𝑏+𝑧

𝑧−𝑏

𝑏∈C

3

𝑏+𝑧

𝑏𝑧 + 𝑐𝑏 ̸= 04

𝑏𝑧 + 𝑐𝑐𝑏𝑧 + 1−𝑏(𝑧 − 𝑐)/𝑏(︀)︀
𝑐log𝑏 𝑧 − 1−𝑏5𝑏𝑧

tet𝑏 (𝑧)6𝑧𝑏7−𝑎2 /𝑧exp(𝑏𝑧 )(︁ 𝜋 )︁
𝑎 tan
𝑧
2
1−𝑐
1 − 𝑐𝑧

8

9

𝑧
𝑐+𝑧
𝑧
1+𝑧

ate𝑏 (𝑧)(︀)︀log𝑏 ln(𝑧)2arctan(𝑧/𝑎)𝜋(︁
1 − 𝑐 )︁log𝑐 1 −𝑧comment𝑏 ̸= 0, 𝑏 ̸= 1, [87][54, 61, 80](4.19), 𝑏 > 0, 𝑏 ̸= −1

𝑎>0
𝑐 ̸= 0 , 𝑐 ̸= 1

1/𝑧

1/𝑧

𝑧
𝑓 = 𝑔 ; 𝑇 𝑛 (𝑧) = 1+𝑛𝑧

𝑏 ̸= 010ln(𝑏+e𝑧 )ln(𝑏𝑧)

e𝑧 /𝑏

11

𝑎𝑧 1/𝑏13(𝑎𝑏 +𝑧 𝑏 )1/𝑏√
2𝑧 1−𝑧 2√
2𝑧 1+𝑧 2(𝑧/𝑎)𝑏(︀)︀log2 arcsin(𝑧)/𝜋
√
(︀ (︀
)︀ )︀
log2 ln 𝑧 + 𝑧 2 +1 /𝜋

14

2𝑧 2 − 1cos(𝜋2𝑧 )15

2𝑧 2 − 1cosh(𝜋2𝑧 )162𝑧/(1−𝑧 2 )tan(2𝑧 )17

2𝑧/(1+𝑧 2 )tanh(2𝑧 )18

Factorial(𝑧)SuFac(𝑧)19𝑏 𝑧 (1−𝑧)

LogisticSequence𝑏 (𝑧) LogisticSequence−1
𝑏 (𝑧)

20

Doya𝑡 (𝑧)Tania(𝑡𝑧)(𝑧 + ln(𝑧) − 1)/𝑡

(5.11), (5.3)

21

Keller𝑡 (𝑧)Shoka(𝑡𝑧)

ArcShoka(𝑧)/𝑡(5.14), (5.18)

22

sin(𝑧)SuSin(𝑡𝑧)

AuSin(𝑧)/𝑡(12.8), [92]23

zex(𝑧) = 𝑧 exp(𝑧) SuZex(𝑡𝑧)AuZex(𝑧)/𝑡(11.1) [89]24

tra(𝑧) = 𝑧 +e𝑧SuTra(𝑡𝑧)

AuTra(𝑧)/𝑡(20.1) [89]𝑃 (𝑇 (𝑄(𝑧)))𝑃 (𝑓 (𝑧))𝑔(𝑄(𝑧))𝑃 (𝑄(𝑧)) = 𝑧12sin(𝜋2𝑧 )sinh(2𝑧 )𝑎 > 0, 𝑏 ̸= 0

log2 (arccos(𝑧))
√
(︀ (︀
)︀ )︀
log2 ln 𝑧 + 𝑧 2 −1 /𝜋
log2 (arctan(𝑧))
(︂ (︂
)︂)︂
𝑧 +1
log2 2 ln𝑧 −1
AuFac(𝑧)30(8.11), (8.19); [66](7.8), (7.19); [70]

In order to get the table of integrals, one builds-up the of table of derivatives: Just take any short combination of basic elementary functions anddifferentiates it. If the result can be simplified to fit the width of thecolumn of the table, then it is declared as a “function that had been
analytically integrated”.
In the similar way, one can deal with superfunctions. Any elementaryfunction 𝑓 , for which the inverse function 𝑔 = 𝑓 −1 also can be expressed
as elementary function, can be taken as an example, if both expressions
for 𝑓 and 𝑔 fit the width of the column of the table. Then, expression𝑡(𝑧) = 𝑓 (1 + 𝑔(𝑧))(3.1)

should be considered. Sometimes, after simplification, this expression
fits the width of the column of the table. Then, this 𝑡 can be declared as“transfer function, for which the superfunction can be expressed analytically”, id est, also in terms of special functions. So, this 𝑓 is declared as
its superfunction, and 𝑔 is declared as corresponding abelfunction. The
most of table 3.1 is built-up in such a way.
Not all transfer functions can be expressed as elementary functions, andnot all superfunctions can be expressed through special functions. And
not all abelfunctions. In this Book, I describe methods, how to deal
with these cases. In general, any holomorphic function can be treatedas transfer function, and the only edges of the range of holomorphism
may limit the construction of the corresponding superfunction and theabelfunction. If the superfunction and the abelfunction are constructed,
supplied with the specific names, described and implemented, they canbe treated as a special functions. I call any function as “special function”,
if (and only if) the properties are revealed, described and the algorithmof the precise evaluation is supplied.
Aiming the application in physics (and, perhaps, in other sciences), I
am interested, first, in those functions, that can be evaluated quickly.
These functions can be used to construct new functions, and used todescribe various phenomena. One can use them in the similar way, as
one use other special functions (sin, bessel, erfc, etc.) These functions
appear in the Table 3.1 with names; I supply also the number of formulaor the cite, to indicate, where the function is described, where can one
find the algorithm for the evaluation.
Table 3.1 appears as the basic toolbox for evaluation of non-integer iteration. First, it worth to check properties of elementary superfunctions.

31

2Construction of elementary superfunctions

As it is mentioned above, to construct Table 3.1, I begin not with atransfer function, but with the superfunction and the abelfunction, applying formula (3.1). As an example, I show, how the 12th line of theTable 3.1 can be verified in language Mathematica:
f[z_] = Sin[Pi 2^z]g[z_] = Log[ArcSin[z]/Pi]/Log[2]f[g[z]]
T[z_]=2 z Sqrt[1 - z^2]
Simplify[T[z] - f[1 + g[z]]]In Mathematica, the argument of function should appear in squaredbrackets. No other trick, specific for Mathematica, is used; the verification can be performed in other languages too.
Table 3.1 collects the simplest (already described) superfunctions. They
can be modified, using the last row of the Table. Any pair of mutually
inverse holomorphic functions 𝑃 and 𝑄 determines the transform, thatcan be applied to any of previous rows, giving the new transfer function
with corresponding superfunction and abelfunction.
In general, the superfunction is not unique (In the similar way, the indefinite integrals are not unique). A special case of transformation of superfunction is displacement of its argument for a constant. In some cases, it
is difficult to recognize this shift. Superfunctions in 14th and 15th rowsof the table correspond to the same transfer function 𝑇 (𝑧) = 2𝑧 2−1; thesesuperfunctions are related with translation of the argument for constanti𝜋 ln(2)/2. Similar relations take place also for other superfunctions; inparticular, for superfunctions of exp𝑏 for 1 < 𝑏 < exp(1/e) , discussed in[61]; this case is mentioned in Chapter 9.
In addition, the scaling transform relates the quadratic transfer function
at rows 14 and 15 of the table with logistic operator (also quadratictransfer function) in raw 19, while parameter 𝑏 = 4. (The superfunctions
can be constructed also for other values of 𝑏, see Chapter 7.)
I invite the Reader to add any new rows to the table, following thetrick above: choose some special function 𝑓 , for which 𝑔 = 𝑓 −1 is also
implemented, and try to simplify 𝑡, determined by equation (3.1).
The main content of this chapter is Table 3.1. It adjusts notations forthe specific superfunctions, described in this Book. Some elementarysuperfunctions are considered with more details in the next chapter.

32

Chapter 4
Elementary superfunctions
In order to use superfunctions, described inthis chapter, the Reader has no need to know,that they are superfunctions. In the similarway, during 40 years, Joudrain has no need toknow, that he speaks in prose [2]. However,many properties of elementary superfunctions
are the same, as properties of other, nontrivial
superfunctions, that cannot be easily expressed
through elementary functions. In order to showthese properties, in this chapter I consider elementary superfunctions.
I would not like the colleagues to say, that theformalism of superfunctions is too complicated[76]. So I begin with very simple example, withlinear function.

1

Fig.

4.1:

These

M.

forty

Jourdain:

years

now,

I’ve been speaking in prose
without knowing it! [2]Iteration of linear function

Consider the linear transfer function

𝑇 (𝑧) = 𝐴 + 𝐵𝑧

(4.1)

where 𝐴 and 𝐵 are constants. For 𝐴 = 1, 𝐵 = 2 , iterates of this function
are shown in figure 4.2. Graphic 𝑦 = 𝑇 𝑛 (𝑥) is plotted versus 𝑥 for various
values of the number 𝑛 of the iterate.
The 𝑛th iterate of function 𝑇 can be written as follows:

𝐵𝑛 − 1
𝑇 (𝑧) = 𝐴
+ 𝐵 𝑛𝑧
𝐵−1
𝑛

33

(4.2)

4

𝑛=
𝑛 = 1 .2𝑛= 1
𝑛 = 0 .8𝑛 = 0 .60𝑛 = .40𝑛 = .2
0

𝑛→∞
𝑛=4
𝑛=3
𝑛=2

𝑦

0.2
−
𝑛 = 0.4−=
𝑛 −0.6𝑛 = −0.8
𝑛 = −1
𝑛=

3
2
1
−4

−3

−2

−1

00

1

−1

2

3

2
𝑛=−
𝑥
4
𝑛 = −4𝑛 → −∞−2−3−4

https://mizugadro.mydns.jp/t/index.php/File:Itelin125T.jpg
Fig. 4.2:

Iterates of linear function (4.1) at 𝐴 = 1, 𝐵 = 2 ; 𝑦 = 𝑇 𝑛 (𝑥)versus 𝑥 for various 𝑛This representation is used to plot figure 4.2. The graphics are straightlines. They cross at the point (𝐿, 𝐿) of the coordinate plane. The fixed
point 𝐿 is determined by the equation 𝐴 + 𝐵𝐿 = 𝐿, that gives

𝐿 = 𝐴/(1 − 𝐵)

(4.3)

At 𝐵 → 1, the fixed point runs to infinity, and the graphics in the analogyof figure 4.2 become parallel. For 𝐴 = 1, 𝐵 = 2 , this is case of figure 4.2,
we get value 𝐿 = −1; so, lines in figure 4.2 cross in point (−1, −1).
Representation (4.2) can be obtained with the general formula (2.14)
with superfunction

1 − 𝐵𝑧𝐹 (𝑧) = 𝐴
1−𝐵
34

(4.4)

and abelfunction(︁𝐵 − 1 )︁𝐺(𝑧) = log𝑏 1 +
𝑧
(4.5)
𝐴
In the special case 𝐵 = 1, the transfer function has no fixed points andrepresentations (4.2), (4.4), (4.5) cannot be used. For this case, id est,
for 𝑇 (𝑧) = 𝐴 + 𝑧 , superfunctions and abelfunctions can be written as
follows:
𝐹 (𝑧) = 𝐴𝑧 , 𝐺(𝑧) = 𝑧/𝐴(4.6)

Their combination gives

𝑇 𝑛 (𝑧) = 𝐹 (1 + 𝐺(𝑧)) = 𝐴 (𝑛 + 𝑧/𝐴) = 𝐴𝑛 + 𝑧(4.7)

Those, who still think, that the formalism is too complicated [76], are invited to check the deduction above and verify, that this case corresponds
to the level of a junior high school.
To iterate the linear function (4.1), theuse of superfunction (4.4) and abelfunction (4.5) can be qualified with term“use sledgehammer to crack a nut”, as
it is shown in figure 4.3. However, thisexample is important: it is simple and it
shows, how the superfunction and abelfunction can be used together to express
Fig. 4.3: Mathematician uses
iterates with formula (4.7).
sledgehammer to crack a nut

2

Rational function

The linear fraction, or “rational function” can be considered as generalisation of the linear function. Let

𝑇 (𝑧) =

𝑈 +𝑉𝑧
𝑊 +𝑧

(4.8)

where 𝑈 , 𝑉 and 𝑊 are constant parameters. First, as an example,
consider function

𝑇 (𝑧) = −1/𝑧

(4.9)

This example corresponds to 𝑈 = −1, 𝑉 = 0, 𝑊 = 0 in formula (4.8).
Negative value of 𝑈 is chosen in order to have positive derivative at the
35

𝑦
8

𝑢=0

6
4
2
0

𝑣 = 0.1

𝑣 = 0.2

𝑢 = −0.1

𝑢 = 0.1
𝑣 = 0.2𝑣 =0

𝑢 = −0.2
𝑣 =0

−2
−4

𝑣 = −0.2
𝑢=0−6

𝑣 = −0.1

−8
−8

−6

−4

−2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:F1xmapT.jpg
Fig. 4.4:

𝑢+i𝑣 = 𝑇 (𝑥+i𝑦) = 1/(𝑥+i𝑦)

positive values of the argument; iterates of a growing function are easy tointerpret in terms of modelling of physical process. Iterates of decreasing
function, contrary, imply dealing with complex values. I am interested
mainly in the applications for physics; so, I consider mainly the growing
transfer functions and growing real-holomorphic superfunctions.
Compex map of function 𝑇 by equation (4.9) is shown in figure 4.4.
For this function, the levels of constant real part and levels of constantimaginary part are circles, and all these circles pass through the originof coordinates.
For real values of the argument, iterates of function 𝑇 by (4.9) are shownin figure 4.5. Lines 𝑦 = 𝑇 𝑛 (𝑥) are plotted versus 𝑥 for various values of𝑛. Below I describe, how these iterates are calculated.

36

𝑛=

6

3

1.5

−4

0
𝑛=

1.

0 .1

4

𝑛=

𝑛=

7
1.

𝑛 = 0.5
𝑛 = 0.3
𝑛 = 0.2𝑦𝑛=
1.4
𝑛 = .3𝑛 = 11.2
𝑛=
𝑛=1
−3
−2
.5
0
=
𝑛

1
−0.
=
𝑛

2

.2
𝑛 = −0𝑛 = −0.3
𝑛 = −0.4
𝑛 = −0.6

1
−1

0
0

1

2

−1
−2

3

𝑛 = −1 𝑥
𝑛 = −1.2
𝑛 = −1.4
𝑛 = −1.5
−
𝑛=

1.6

−3
−4
𝑛=

7
1.
−

https://mizugadro.mydns.jp/t/index.php/File:Frac1zt.jpg
Fig. 4.5:

𝑦 = 𝑇 𝑛 (𝑥) for 𝑇 (𝑧) = −1/𝑧 by formula (4.12) at various 𝑛(︁ 𝜋 )︁
𝑧
𝐹 (𝑧) = tan
2
𝐺(𝑧) = 𝐹 −1 (𝑧) =(4.10)

2arctan (𝑧)𝜋(4.11)

Then, the 𝑛th iterate of the transfer function 𝑇 𝑛 (𝑧) = 𝐹 (𝑛 + 𝐺(𝑧))appears to be(︁ 𝜋 )︁
𝑛 𝑧−1 − cot𝑛2(︁ 𝜋 )︁
𝑇 (𝑧) =
− cot
𝑛 +𝑧
2

(4.12)

This case is represented in row 7 of table 3.1. Expression (4.12) is used
to plot figure 4.5.
37

0

−0.
1

𝑛=

𝑛=

𝑛 = −0.5
𝑛 = −0.4
𝑛 = −0.3
𝑛 = −0.2

5𝑛 = −1

𝑦

𝑛=2

𝑛=1

4
3

𝑛 = 0 .2
𝑛 = 0 .3
𝑛 = 0.54
𝑛 = 0.
𝑛=1
𝑛=2

2
1
1
2
3
𝑛 = −2𝑛 = −1

4

5

𝑥

𝑛=

−3

𝑛=3

0.1
=
𝑛

https://mizugadro.mydns.jp/t/index.php/File:Fracit05t150.jpg
𝑛Fig. 4.6: 𝑦 = 𝑡 (𝑥) by (4.13) at 𝑐 = 0.5

Consider one more special case of formula (4.8). Let𝑧𝑐+𝑧(4.13)

𝑧
1−𝑐𝑛
𝑛
𝑐 +
𝑧
1−𝑐

(4.14)

𝑡(𝑧) =
where 𝑐 is constant. Then𝑡𝑛 (𝑧) =Iterates of function 𝑡 for 𝑐 = 0.5 are shown in figure 4.6. The same for𝑐 = 1 are shown in figure 4.7, and the same for 𝑐 = 2 are shown in figure
4.8. Below I show, how these iterates are evaluated.
For the transfer function 𝑡 by (4.13), superfunction 𝑓 can be written as
38

0
𝑛=

5

𝑛 = −0.5
𝑛 = −0.4
𝑛 = −0
.3
𝑛=−
0.2
𝑛=
−0
.1

𝑛 = −1

𝑦

4
0
=
𝑛

3

2
𝑛 = 0.
𝑛 = 0 .3
𝑛 = 0 .4
𝑛 = 0.5

2

𝑛
𝑛= =2
3

.1

1

𝑛=1
𝑛=2
𝑛=

0
0

−
4

4
=
𝑛

1
2
𝑛 = −3𝑛 = −2345𝑥

https://mizugadro.mydns.jp/t/index.php/File:Fracit10t150.jpg
𝑛Fig. 4.7: 𝑦 = 𝑡 (𝑥) by (4.13) at 𝑐 = 1

follows:

𝑓 (𝑧) =

𝑐−1
𝑐𝑧 − 𝑐(4.15)

Corresponding Abel function 𝑔 = 𝑓 −1 is𝑔(𝑧) = log𝑐(︁𝑐−1 )︁
1+𝑧(4.16)

Expressions (4.15) and (4.16) are used to evaluate 𝑡𝑛 (𝑧) = 𝑓 (𝑛+𝐺(𝑧))and plot figures, in particular, figures 4.6, 4.7 and 4.8. The Reader isinvited to plot iterates of function 𝑡 by formula (4.15) at other values of𝑐 too.
39

𝑛=

0

−0
.1
𝑛=

𝑛 = −0.5
𝑛 = −0.4
𝑛 = −0
.3
𝑛=−
0.2

5

𝑛 = −1

𝑦

4
𝑛

3

.2
𝑛=0
𝑛 = 0 .3
𝑛 = 0 .4
𝑛 = 0 .5

2
1
0
0

.1
0
=

𝑛=1
𝑛=2
1

2

3

4

5

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Fracit20t150.jpg
Fig. 4.8:

𝑦 = 𝑡𝑛 (𝑥) by (4.13) at 𝑐 = 2

The iterates of the rational function of real argument show the smoothtransition from function 𝑡 to the identity function and then to inverse
function 𝑡−1 . All the curves in figures 4.6, 4.7 and 4.8 pass through thefixed point. This value is mapped to itself at the iterates of the transfer
function. This property is not specific feature of the rational function.
Other functions, considered on the following chapters, have the same
property.
For construction and uniqueness of superfunctions in the following sections, it is important to consider them in the complex plane. The complex maps help to understand properties of these functions. Following

40

𝑦

𝑣 =0

𝑣 =0

𝑣 = −0.5

𝑢=0

8
6

𝑣 =0

4

𝑢 = −0.12

𝑢 = −0.5

0

𝑣 =0

𝑢=0

𝑣 = −0.1
𝑢 = 0.1 𝑣 = 0
𝑣 = 0.1−2

𝑢 = −0.5

−4

𝑢=0

𝑣 =0

−6

𝑢 = −0.5

𝑢=0

𝑣 =0

𝑣 =0

−8
−8

−6

−4

−2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Fz2z1mapT.jpg
Fig. 4.9:

𝑢+i𝑣 = 𝑓 (𝑥+i𝑦) by (4.15) at 𝑐 = 2

the idea to begin with simple examples, I suggest the complex mapsof superfunction 𝑓 and abelfunction 𝑔 . For 𝑐 = 2, the complex map ofsuperfunction 𝑓 by formula (4.15) is shown in figure 4.9. Similar mapof the Abel function 𝑔 is shown in figure 4.10.
Function 𝑓 by formula (4.15) is periodic; its period

𝑃 = 2𝜋i/ ln(𝑐)(4.17)At 𝑐 = 2, this period 𝑃 = 2𝜋i/ ln(2) ≈ 9.06472 i. Vertical size of figure 4.9
covers a little bit more that two periods of this function. For real 𝑐, theperiod is pure imaginary; the map reproduces itself at the translations
for integer factor of |𝑃 | along the imaginary axis.
The inverse function 𝑔 = 𝑓 −1 is expressed by formula (4.16). It is Abel
41

𝑦

𝑢=0

𝑣 = −0.1

𝑣 = −0.1

8
6
4
2
0
−2

𝑣 =0
𝑢 = −0.2

𝑣 =0
𝑢 = 0.2

−4
−6
−8

−8

−6

𝑢 = 0.1

𝑣 = 0.2
𝑢=0

𝑢 = −0.1
𝑣 = 0.1
−4

−2

0

2

𝑣 = 0.1468𝑥

https://mizugadro.mydns.jp/t/index.php/File:Gz2z1mapT.jpg
Fig. 4.10:

𝑢+i𝑣 = 𝑔(𝑥+i𝑦)by (4.16) at 𝑐 = 2function for the transfer function 𝑡. Abelfunction 𝑔 is shown with its
complex map in figure 4.10 for the same value of parameter, 𝑐 = 2, asfunction 𝑓 in figure 4.9
Function 𝑔 has two branch points, 𝑐 − 1 and zero. For real 𝑐, the cut
line between these two points belongs to the real axis. Equilnes are
symmetric with respect to reflections from line ℜ(𝑥) = (𝑐−1)/2.
With superfunction 𝑓 and abelfunction 𝑔 by formulas (4.15) and (4.16),
iterates of the transfer function 𝑡 can be written as usually,𝑡𝑛 (𝑧) = 𝑓 (𝑛 + 𝑔(𝑧))(4.18)The readers are invited to check, that this representation agree withexpression (4.14). It is better to do this exercise for the simple. Then it
will be easier to reproduce the similar exercises for other superfunctionsin the following sections.
42

𝑦
4
3
2

𝑣 =1
0
𝑣 =5
𝑣 =0
𝑣 = −5

1
0
−1

10
𝑢= 5
𝑢= 0
𝑢=

−2

𝑢=
𝑢 = −5
−1
0

−3
−4
−5
−5

−4

−3

−2

−1

0

1

2

3

4

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Z2itmapT.jpg
Fig. 4.11:

3

𝑢+i𝑣 = 𝑇 (𝑥+i𝑦) = (𝑥+i𝑦)2 , by (4.19) for 𝑎 = 2

Power function and its iterates

Consider the power function(4.19)

𝑇 (𝑧) = Pow𝑎 (𝑧) = 𝑧 𝑎

For 𝑎 = 2, the map of function 𝑇 is shown in figure
4.11. Here, I suggest the superfunction of the powerfunction, id est, the superpower function. (However,Fig. 4.12: Superpowerterm “superpower” may have also other meaning, as
it is shown in figure 4.12, but such a meaning is not used in this Book).
Consider iterates of the power function 𝑇 by (4.19), they can be written

43

𝑛=

0

𝑛=
0
𝑛 = .6
0.
𝑛= 5
0.4
𝑛=
0 .3
𝑛=
0 .2
𝑛=
0 .1

𝑛→ ∞
𝑛=3
𝑛=2

9

𝑛=1

𝑛

𝑥2

8

𝑛=

.1
0
−

0.2
−
𝑛= 3
0.
−
𝑛 = .4
−0
=
𝑛
5
−0.
=
𝑛
0.6
−
=
𝑛

7
6
5
4

𝑛 = −13
2

𝑛 = −2𝑛 = −31𝑛 → −∞

1

2

3

4

5

6

7

8

9

𝑥

https://mizugadro.mydns.jp/t/index.php/File:IterPowPlotT.jpg
Fig. 4.13:

𝑦 = 𝑇 𝑛 (𝑥) = Pow2 𝑛 (𝑥) для различных 𝑛.

as follows:𝑛𝑛

𝑇 (𝑧) = Pow𝑎 (𝑧) = 𝑧

𝑎𝑛

(︁

)︀)︁
= Pow𝑎 Pow𝑎 ...Pow𝑎 (𝑧)..
⏟⏞𝑛 evaluations of function Pow𝑎(︀

(4.20)

Figure 4.13 shows 𝑇 𝑛 by formula (4.20) for 𝑎 = 2 at various 𝑛. Iteratesof the power function can be expressed also with the general formula,)︁𝑇 (𝑥) = 𝐹 𝑛 + 𝐺(𝑥)𝑛(︁

(4.21)

where 𝐹 is superfunction of the transfer function 𝑇 and 𝐺 = 𝐹 −1 is theabelfunction.
For the power function, the superfunction (id est, the superpower func-

44

tion) can be written as follows:

(︁

)︁
(︀
)︀
𝐹 (𝑧) = exp exp(ln(𝑎) 𝑧) = exp2 ln(𝑎) 𝑧(4.22)

Inverting this representation, one can get the abelfunction; id est, the“abelpower” function,(︁)︁𝐺(𝑧) = ln ln(𝑧) / ln(𝑎) = ln2 (𝑧)/ ln(𝑎)(4.23)

These formulas correspond to the 6th row of table 3.1.
For the transfer function 𝑇 by (4.19), iteration (10.16) can be simplified,𝑛

𝑇 (𝑧) = exp

2

(︁

(︀
)︀)︁
𝑛
2
ln(𝑎) 𝑛 + ln (𝑧)/ ln(𝑎) = 𝑧 𝑎(4.24)

leading to the 3d expression in equation (4.20).
In such a way, iterates of the transfer function 𝑇 by (4.19) appear to befunction of the similar kind (also power function). I invite the Reader toplot the complex maps of superpower function 𝐹 by (4.22), abelpowerfunction 𝐺 by (4.23) and iterates of power function by (4.24).
For 𝑇 (𝑧) = 𝑧 𝑎 , the simple relations take place:

𝑇 (𝑧 𝑏 ) = 𝑇 (𝑧)𝑏(4.25)At 𝑏 = 𝑎 = 2, in addition, the following relation hakes place:𝑇 𝑏 (𝑧) = 𝑇 (𝑧)𝑏(4.26)

There is common confusion (the wrong public opinion), that some simple equivalents of relations (4.25), (4.26) should take place for othervalues of 𝑏 and for other functions too. In particular, one of critics ofthe publication about half iteration of factorial [62] had insisted, that
Factorial1/2 (𝑧) = Factorial(𝑧)1/2 , and√I was not successful explaining,that it is not the case, even if notation Factorial = Factorial1/2 is used.
The factorial is considered in this Book in the Chapter 8. However, before to deal with factorial, I would like to consider simple example. The
power function is one of the simple examples. I invite the Reader tocheck, that even for 𝑎 = 3, relation (4.26) is not valid (except of somespecific values of 𝑧 ).
Having the explicit representation, one can calculate all the iterates necessary, including the non-integer iterates. But not so many elementaryfunctions are inverse of some other elementary functions, Perhaps, it
is better to refer to special functions. The simple example of the nonelementary function is considered in the next chapter.
45

Chapter 5
Tania and Shoka
𝑦
3
2
1
−3

−2

−1

0

𝑥)
(
ka
o
)
Sh ia(𝑥𝑦 = Tan
𝑦=

1

2

3

𝑥

https://mizugadro.mydns.jp/t/index.php/File:ShokotaniaT.png
Fig. 5.1: Functions Tania and Shoka by (5.1) and (5.2)

This chapter considers functions, that have applications in laser science.
I call them Tania and Shoka. For some vicinity of the real axis, thesefunctions can be defined as follows:

Shoka(𝑧) = 𝑧 + ln(e−𝑧 +e−1)(5.1)

Tania(𝑧) = WrightOmega(𝑧 +1)(5.2)

At |ℑ(𝑧)| ≥ 𝜋 , function WrightOmega(𝑧) behaves in a way I dislike.
To avoid confusions, for the function I like, I use name Tania. Below I
define functions Shoka and Tania for the complex argument. However,
the representations (5.1) and (5.2) are sufficient to plot Shoka and Tania
in figure 5.1.
This chapter retells the contents of articles [85, 86]. This chapter shouldbe especially important for the narrow specialists, who works in the laser
science.

46

𝑦

𝑣 =6
𝑣 =5
𝑣 =4
𝑣 =3
𝑣 =2
𝑣 =1
𝑣 =0

6

2

𝑢=0

0

𝑣 =0

−2

𝑢=0

−𝜋
−4

𝑢=2

cut

𝑢=1

4
𝜋

𝑣 = −1𝑣 = −2𝑣 = −3𝑣 = −4𝑣 = −5𝑣 = −6

cut

−6
−6

−4

−2

0

2

4

6

𝑥

https://mizugadro.mydns.jp/t/index.php/File:TaniaContourPlot100.png
Fig. 5.2: Complex map of function Tania,

1

𝑢+i𝑣 = Tania(𝑥+i𝑦)

Tania and Arctania

Let function Tania be solution 𝑓 of the differential equation𝑓 ′ (𝑧) =𝑓 (𝑧)
1 + 𝑓 (𝑧)(5.3)

with additional condition 𝑓 (0) = 1, where contour of integration of (5.3)
for 𝑓 (𝑧) goes first from zero to imaginary part of 𝑧 along the imaginary
axis, and then, along the line, parallel to the real axis, goes to point 𝑧 .
Figure 5.1 shows function Tania of real argument. For moderate valuesof imaginary part of the argument, solution 𝑓 = Tania of equation (5.3)
is expressed through the special function WrightOmega with equation(5.2). Complex map of function Tania is shown in figure 5.2.
47

𝑦

𝑣 =8
𝑣 =7
𝑣 =6
𝑣 =5
𝑣 =4
𝑣 =3
𝑣 =2
𝑣 =1
𝑣 =0
𝑣 = −1𝑣 = −2𝑣 = −3𝑣 = −4𝑣 = −5𝑣 = −6𝑣 = −7𝑣 = −8

6
4
2
cut

0
−2

𝑢=0
𝑢=1
𝑢=2
𝑢=3
𝑢=4

−4
−6
−6

−4

−2

0

2

4

6

𝑥

https://mizugadro.mydns.jp/t/index.php/File:ArcTaniaMap.png
Fig. 5.3:

𝑢+i𝑣 = ArcTania(𝑥+i𝑦)

For real values of argument, Tania is positive and grows monotonously.
Toward the negative values of the argument, Tania decays exponentially.
At zero, Tania grows with tangent 1/2. At large positive values of theargument, Tania grows in a way, similar to the linear function with
tangent unity.
Function Tania, shown in figure 5.2, has two branch points, −2 ± i𝜋 .
The cut lines are directed parallel to the real axis toward its negative
direction; they are determined by specification of path of integration ofequation (5.3).
Complex map of the inverse function, id est, ArcTania = Tania−1 is48

shown in figure 5.3. It can be expressed as elementary function,

ArcTania(𝑧) = 𝑧 + ln(𝑧) − 1(5.4)

There are no complex constants in the representations of functions Tania
and Arctania; these functions are real-holomorphic:

Tania(𝑧 * ) = Tania(𝑧)* , ArcTania(𝑧 * ) = ArcTania(𝑧)*(5.5)

Functions Tania and ArcTania look similar to the linear function at large
values of the argument; the lines of levels of constant real part and thoseof constant imaginary part form almost rectangular grid. Functions
ArcTania also have almost linear asymptotic.
The almost linear asymptotic behaviour of Tania(𝑧) at large |𝑧| ≫ 1
holds for the most of the complex plane, except the strip ℜ(𝑧) < 0,|ℑ(𝑧)| ≤ 𝜋 . In the strip |ℑ(𝑧)| < 𝜋 , at ℜ(𝑧) → −∞, function Tania(𝑧)decays exponentially, this agrees with graphic of this function for realargument at figure 5.1.
For moderate values of the imaginary part of the argument, function
Tania can be expressed through the known special function WrightOmega[12, 117] with equation (5.2). In particular, one has no need to makeany difference between Tania(𝑧) and WrightOmega(𝑧 +1) for real 𝑧 .
For the efficient evaluation of Tania, its asymptotic expansions can beused. The whole complex plane can be covered with these expansions.
At large values of the argument, Tania can be expanded as follows:

ln(𝑧)−1 ln(𝑧)2 −4 ln(𝑧)+3++ .. (5.6)
𝑧
2𝑧 2The effective small parameter of the expansion (5.6) is ln(𝑧)/𝑧 . For
negative values of ℜ(𝑧), this expansion is valid while |ℑ(𝑧)| > 𝜋 However,this representation does not work between the cut lines at figure 5.2. For
the representation of Tania in this half-strip, I define the new variable
𝜀 = exp(1+𝑧); then Tania can be expanded as follows:
Tania(𝑧) = 𝑧 + 1−ln(𝑧) +

3
8
125 5
Tania(𝑧) = 𝜀 − 𝜀2 + 𝜀3 − 𝜀4 +𝜀 + 𝑂(𝜀6 )
2
3
24

(5.7)

The expansions above are not good for the moderate values of the argument, and especially poor is the approximation in vicinity of the branch
points. For the branch point, the following expansion takes place:33
9 5
Tania(𝑧) = −1 + 3𝑡 − 3𝑡2 + 𝑡3 + 𝑡4 +𝑡 + ..
4
10
160
49

(5.8)

√︂

)︀
2 (︀
𝑧 +2−𝜋i .
9In addition, one may use the Taylor expansion at zero:
where 𝑡 = i

𝑧 𝑧2
𝑧3
𝑧4
13𝑧 5
47𝑧 6
Tania(𝑧) = 1 + +
−
−
+
−
+ ..
2 16 192 3072 61440 1474560(5.9)

With the representations above, for every point of the complex plane,
for Tania(𝑧), one can get the “zeroth” approximation, let it be called 𝑠0 ,with few correct significant figures. Then, in order to get the maximal
precision for the “complex double” variable, it is sufficient to make threeor four iterations by the Newton method

𝑠𝑛+1 = 𝑠𝑛 +𝑧 − ArcTania(𝑠𝑛 )ArcTania′ (𝑠𝑛 )(5.10)

where ArcTania′ (𝑧) = 1+1/𝑧 is derivative of ArcTania. I remind that,
ArcTania by equation (5.4) is elementary function. In such a way, Tania
can be evaluated quickly and precisely. The C++ implementation offunction Tania with this algorithm for “complex double” variables isloaded as https://mizugadro.mydns.jp/t/index.php/Tania.cin
Function Tania is simpler than function WrightOmega. If necessary,
Tania can be used to evaluate WrightOmega.
Function Tania has simple physical meaning. It represents dependence
of intensity of light on the length of its propagation in laser with simple
model of the active medium. The argument has sense of coordinate measured in units of the inverse increment of the low signal. The returned
value has sense of intensity, measured in the units of saturation.
In this Book, function Tania is used many times. In thischapter, Tania appears as superfunction of the special
function Doya, considered in the nest section. I use thetrick, mentioned in the chapter 3: First, I choose superfunction, id est, Tania, and then, I construct the transfer
function for it; I call this transfer function “Doya”. I am
grateful to Valérie Doya (Figure 5.4), she kindly allowed
to use her name for the function, that appeared during ourcollaboration at Nice in 2010.

50

Fig.

5.4:

V.Doya[118]𝑡3
𝑣 =3 3 𝑣 =𝑣 =2 2 𝑣 =2
𝑣 =1 1 𝑣 =1
𝑣 =0 0 𝑣 =0
𝑣 = −1 −1 𝑣 = −1
𝑣 = −2 −2 𝑣 = −2𝑣 = −3 −3 𝑣 = −3

−1
−2
−3

𝑢=1

0

𝑢=

1

0

𝑢=1
𝑢=2
𝑢=3

𝑢=0

2

𝑢 = −13

𝑢 = −2𝑡𝑣 =0𝑥−3 −2 −1 0
1
2
3
−3 −2 −1 0
123
https://mizugadro.mydns.jp/t/index.php/File:Doya10map4bT60.png
https://mizugadro.mydns.jp/t/index.php/File:Doyam10map4bT100.png
Fig. 5.5:2𝑥

𝑢+i𝑣 = Doya(𝑥+i𝑦), left, and 𝑢+i𝑣 = Doya−1 (𝑥+i𝑦), at rightTransfer function DoyaFunctions Tania and ArcTania from the previous section allow to buildup the “solvable” transfer function. I call it Doya,(︀)︀
𝑇 (𝑧) = Doya(𝑧) = Tania 1 + ArcTania(𝑧)(5.11)

Complex map of function Doya, and also map of its inverse function

ArcDoya = Doya−1 are shown in figure 5.5. At large values of theargument, each of these functions looks similar to identity function, butthey have the branch points and cuts in the central part of the maps.
While functions Tania and ArcTania are already implemented, the evaluation of function Doya is straightforward. In addition, in vicinity of
the real axis, Doya can be expressed through the known special function
LambertW [113]:(︀)︀
Doya(𝑧) = LambertW 𝑧 e𝑧+1

(5.12)

According to definition (5.11), Tania is superfunction of Doya, and ArcTania is its abelfunction. The 𝑛th iterate of Doya can be written as
follows:(︁)︁Doya (𝑧) = Tania 𝑛 + ArcTania(𝑧)𝑛51(5.13)

0

𝑛=

𝑛=6
𝑛=
𝑛= 5
4
𝑛=
3
𝑛=
2
𝑛=
1

𝑦
3

=
𝑛

1
−

−2
𝑛= 3
−
=
𝑛 4
−
=
𝑛 5
−
=
𝑛

2
1
0

1

2

3

𝑥

https://mizugadro.mydns.jp/t/index.php/File:DoyaPlotT100.png
Fig. 5.6:

𝑦 = Doya𝑛 (𝑥) по формуле (5.13)

These iterates are shown in figure 5.6. The graphics represent 𝑦 =Doya𝑛 (𝑥) versus 𝑥 for various values of the number 𝑛 of iterate.
As it is mentioned above, function Tania has simple physical sense; it
describes evolution of a signal in a simple model of a uniform saturable
amplifier (or absorber) at the appropriate choice of units of length andunits of intensity. Similar sense can be given to function Doya, it appears as dependence to the output intensity of this amplifier on its input
intensity [85, 86].
Functions Doya, Tania and ArcTania give an example, when the transfer
function, the superfunction and the abelfunction can be expressed interms of special functions, already described in the literature of 20th
century. In the following section, one more example of this kind isconsidered.

52

𝑦
cut

𝑢=1

4
2
0

𝑢=2
𝑢=3
𝑢=4
𝑢=5

6

cut

−2
−4
−6

cut

−6

−4

−2

0

2

𝑣 =7
𝑣 =6
𝑣 =5
𝑣 =4
𝑣 =3
𝑣 =2
𝑣 =1
𝑣 =0
𝑣 = −1𝑣 = −2𝑣 = −3𝑣 = −4𝑣 = −5𝑣 = −6𝑣 = −74

6𝑥

https://mizugadro.mydns.jp/t/index.php/File:KellerMapT.png
Fig. 5.7:

3

𝑢+𝑣 = Keller(𝑥+i𝑦) by (5.14).

Keller, Shoka and ArcShoka

Wave packet of light (or any other quasi–classical bosons) in a uniform
amplifier can be characterised with its energy or its fluence; this quantity
can be denoted as “signal”. Roughly, fluence is energy of pulse per areaof its transversal cross-section. In analogy with the continuous-wave
amplifier, the signal at the output of the amplifier can be considered
as function of the input; and this dependence can be interpreted as thetransfer function of the amplifier. For the simple model of the active
medium, this function can be expressed as elementary function. Complex map of this function is down in figure 5.7; I call it “Keller function”
and define it as follows:

53

𝑦
6
4
𝜋

cut

−2

0

𝑢=1
𝑢=2
𝑢=3

𝑢=0

0

𝑢 = −0.4

2

−2
−𝜋
−4

cut

−6
−6

−4

2

𝑣 =7
𝑣 =6
𝑣 =5
𝑣 =4
𝑣 =3
𝑣 =2
𝑣 =1
𝑣 =0
𝑣 = −1𝑣 = −2𝑣 = −3𝑣 = −4𝑣 = −5𝑣 = −6𝑣 = −74

6𝑥

https://mizugadro.mydns.jp/t/index.php/File:KellerMapT.png
Fig. 5.8:

𝑢+𝑣 = ArcKeller(𝑥+i𝑦) by (5.16).

(︁)︁
Keller(𝑧) = 𝑧 + ln e − e (e − 1)−𝑧

(5.14)

Complex map of this function is shown in figure 5.7.
The inverse function ArcKeller = Keller−1 can be written as follows:(︂

1 e−1 −𝑧
ArcKeller(𝑧) = 𝑧 + ln
+
e
e
e

)︂

(5.15)

Complex map of function ArcKeller is shown in Figure 5.8. Maps offunctions Keller and ArcKeller look similar; the following relation takes
place:

ArcKeller(𝑧) = Keller(𝑧 − i𝜋 − 1) − 1 + i𝜋

54(5.16)

In publications [30, 35] by Ursula Keller (fig.5.9), anotherrepresentation is used,(︁)︁𝑧Keller(𝑧) = ln 1 + e (e − 1)

(5.17)

3

U.Keller

https://mizugadro.mydns.jp/t/index.php/File:KellerDoyaT.png

𝑦

5.9:

K

Function Keller of real argument
is compared to function Doya in
figure 5.10. These functions look
similar.

Fig.

elloy er(a( 𝑥)𝑥)This representation is equivalent of expression (5.14), while|ℑ(𝑧)| < 𝜋 . In particular, representation (5.17) can be usedinstead of (5.14) for real positive values of 𝑧 .

D

𝑦=

𝑦=

Description of pulses usually ismore difficult, than consideration of the stationary regime; forpulses, there is additional parameter, time.
However, forthe simple model above, thepulsed regime is simpler, than
the steady-state case, because,
for the pulses, the transfer function, and the superfunction canbe expressed as elementary functions. In such a way, function Fig. 5.10: Comparison of functions
Shoka is simpler than function Keller and Doya by (5.14) and (5.11)
Tania, and function Keller is simpler than function Doya.

2
1

1

2

𝑥

Iterates of function Keller are shown in figure 5.11. They look similar to
iterates of function Doya shown in figure 5.6. By general formula 2.14,
the iterates of the Keller function can be expressed through its superfunction 𝐹 = Shoka and abelfunction 𝐺 = ArcShoka; these functions canbe expressed as follows:

(︁

)︁
+e−1

(5.18))︂(︂
1−e−𝑧
ArcShoka(𝑧) = 𝑧 + ln
e−1

(5.19)

Shoka(𝑧) = 𝑧 + ln e

−𝑧

55

3

0
𝑛=

𝑛=5
𝑛=
4
𝑛=
3
𝑛=
2
𝑛=
1

𝑦

=
𝑛

2

=
𝑛

1
−
−2

3
−
𝑛=
4
−
= 5
𝑛
𝑛=−

1

𝑥
0https://mizugadro.mydns.jp/t/index.php/File:KellerPlotT.png
1
2
3
Fig. 5.11:

𝑦 = Keller𝑛 (𝑥) .

Complex maps of functions Shoka and ArcShoka are shown in figures
5.12 and 5.13. These maps look similar to those for functions Tania and
ArcTania shown in figures 5.2 and 5.3.
For real values of the argument, functions Tania and Shoka are comparedin figure 5.1 mentioned in the preamble of this chapter. Both functionsin the left hand side of the graphic have the exponential growth withincrement unity; both pass thorough point (0,1) and both grow almost
linearly in the right hand side of the graphics.
Complex maps of functions Shoka and ArcShoka in figures 5.12 and 5.13
look similar. One of them can be obtained from another with constantdisplacement of the argument and addition of some constant to its value.
This can be expressed with relation(︀)︀
ArcShoka(𝑧) = Shoka 𝑧 − i𝜋 − ln(e−1) − ln(e−1) + i𝜋
56(5.20)

4
𝜋

𝑣 =7
𝑣 =6
𝑣 =5
𝑣 =4
𝑣 =3
𝑣 =2
𝑣 =1
𝑣 =0
𝑣 = −1𝑣 = −2𝑣 = −3𝑣 = −4𝑣 = −5𝑣 = −6𝑣 = −7

0

2

6

𝑢=0
cut

2

𝑢=0

0

𝑣 =0

−2

𝑢=0

−𝜋
−4

𝑢=2
𝑢=3
𝑢=4

6

𝑢=0
𝑣 ≈ 2𝜋

𝑢=1

𝑦

cut

𝑢=0

−6 𝑣 ≈ −2𝜋

𝑢=0
−6

−4

−2

4

𝑥

https://mizugadro.mydns.jp/t/index.php/File:ShokaMapT.png
Fig. 5.12: 𝑢+i𝑣 = Shoka(𝑥+i𝑦)

In vicinity of the positive part of the real axis, the map of function
Shoka in figure 5.12 looks similar to the map of function Tania at Figure5.2. However, I cannot suggest any simple expression of function Tania
through function Arctania. I know no analogy of formula (5.20) for
functions Tania and ArcTania.
There are also qualitative differences between Tania and Shoka. Tania
had only two branch points, and accordingly, two cut lines. Shoka has
countable set of branch points and cuts.
All the six functions Tania, ArcTania, Doya, Shoka, ArcShoka and Keller
look similar to the linear function at the large values of the real partof the argument. In the right hand side of the complex maps in figures
5.2, 5.3, 5.5, 5.12 5.13, 5.7, the structure of levels of constant real part

57

𝑦

4
2
0

𝑢=1
𝑢=2
𝑢=3

cut

𝑢=0

6

𝑣 =7
𝑣 =6
𝑣 =5
𝑣 =4
𝑣 =3
𝑣 =2
𝑣 =1
𝑣 =0
𝑣 = −1𝑣 = −2𝑣 = −3𝑣 = −4𝑣 = −5𝑣 = −6𝑣 = −7

0

2

cut

−2
−4
−6

cut

−6

−4

−2

4

6

𝑥

https://mizugadro.mydns.jp/t/index.php/File:ArcShokaMapT.png
Fig. 5.13:

𝑢+i𝑣 = ArcShoka(𝑥+i𝑦).

In order to catch (or to remember) the properties of function Doya, the left hand side part offigure 5.5 is repeated in figure 5.14. The map isshown with 𝑢+i𝑣 = Doya(𝑥+i𝑦), but the picture isrotated for 90o in the positive direction (counterclock wise). The thick lines show levels 𝑢 = −0.4,𝑣 = ±1.2 , 𝑣 = ±1.4 The map looks as a draw of
a Researcher, who has found something extraordinary in a big Book. Perhaps, that Book is about
superfunctions. The Reader is invited to check

58

Fig. 5.14: Eureka!

https://mizugadro.mydns.jp/t/index.php/File:Doya500.png

and levels of constant imaginary part form grid of lines, almost parallelto the real or imaginary axes.

that values of the 3 parameters, indicated above, provide the structure,
shown in figure 5.14. The range of the almost linear transfer appears
above the “head” of the contour of the “human” in the figure.
Similarity of the transfer functions Doya and Keller and, accordingly,
similarity of their superfunctions Tania and Shoka indicate, that foranalysis of the nonlinear media, these functions should be measured withseveral significant figures; over-vice, the measurement will not allow tomake choice between divergent models. At the convenient measurementof the nonlinear response of the medium, variation of the intensity in asample should be small: over-vice, it is difficult to guess, namely which
intensity does the gain or absorption correspond to. At small variation of
the intensity, the precision of its measurement is poor. This difficulty can
be avoided with superfunctions. The transfer function of the opticallythick sample should be measured, and then, the superfunction, id est,
the evolution of intensity inside the sample can be reconstructed. Insuch a way, the number of parameters of the model, can be reduced:
one of parameters, namely, the length of the amplifier, can be excluded
from the model.
My particular interest in superfunctions is related to the ability of the
precise measurement of the gain (or absorption) in the nonlinear medium
versus intensity. The application may refer to investigation of limits ofvalidity of the commonly-used model of the Yb-doped crystals, glasses
and ceramics. The new effects are expected to appear at the edge of thelimit of applicability. One of such effects is described in the article aboutswitching of emissivity of Yb-doped samples, or the “Bisson effect” [48],but namely in that case, the main mechanism of the phenomenon seems
to have thermal origin: the sample warms, and this warming enhances
the absorption (and heating), leading to the avalanche behaviour, that isdifficult to interpret in terms of the transfer function of a single variable,considered in this Book 1 .
1 The name “Bisson effect” or ‘bison effect” refers to the analogy of the avalanche of the tribe ofrunning bison, who awake, mock and force to run the new and new animals. However, the analogy
is limited, because the electrons, that were expected to be popped up into the conduction band,are neither bison, nor even bosons, but fermions.

59

4Overview

The six functions, (Tania, Shoka, ArcTania, ArcShoka, Doya and Keller)
are defined in this chapter. These functions show the realistic examples, the physically-meaningful transfer functions and the corresponding
meaningful superfunctions are expressed in terms of elementary functions. These example will be used in this book later, to illustrate moregeneral methods of construction and evaluation of superfunctions.
My main claim is that I can construct the superfunction, abelfunction
and, therefore, the non-integer iterates of any growing real-holomorphic
function 2 . Even more, everybody, after to read this Book, also can dothe same.
In the following chapters, I consider methods, that can be used for various transfer functions, even if the superfunction cannot be expressed
through the special functions known since century 20. One of thesecases is considered in the next chapter.

2 With the same methods, the non-real, but still holomorphic functions also can be treated; suchan example is considered in chapter 18.
60

Chapter 6
Regular iteration
Regular iteration is way to construct iterates of the transfer function,that are regular in vicinity of its fixed point. The transfer function is
supposed to be holomorphic. In the most of examples, considered inthis Book, the transfer function is assumed to be also real-holomorphic.
In addition, in this chapter, I assume, that the fixed point is real. 1
This Book deals with solutions 𝐹 of the transfer equation (2.12); I repeat
it here(︀)︀𝐹 (𝑧 +1) = 𝑇 𝐹 (𝑧)Assume, that the transfer function 𝑇 is real-holomorphic, and its fixed
point 𝐿 is real, id est,

𝑇 (𝐿) = 𝐿 ,𝐿 = 𝐿*(6.1)

In addition, I assume, that 𝑇 ′ (𝐿) > 0, this case is easier to interpret 2 .
For this case, In this chapter, the specific iterates of the transfer function
are constructed, that can be expanded into convergent Taylor series atleast in vicinity of the fixed point 𝐿. In this sense, they are regular.
On the other hand, the iterates are constructed with definite iteration
procedure, described below, in a regular straightforward way. Hence,
there are at least two independent reasons to use name “regular iteration”
for the procedure below.
1 The same regular iteration can be applied also for the case of non-real fixed point, but theresulting iterates of the transfer function are not real-holomorphic. The real-holomorphic functions
often are easier to interpret (and to apply in Physics), than the more general complex case. This
Book is planned as applied, so, the most of examples here refer to the real-holomorphic functions.
2 Assumption 𝑇 ′ (𝐿) > 0 is natural. Case 𝑇 ′ (𝐿) < 0 is difficult to interpret in terms of real valued
non-integer iterates, because, 𝑇 𝑛 ′ (𝐿) should approach unity at 𝑛 → 0, but, for realistic cases, should
avoid zero. Practically, this means that the non-integer iterates are complex, not real, as in thecase 𝑇 ′ (𝐿) > 0

61

1

General formula

Assume that 𝐿 is fixed point of the transfer function 𝑇 , id est, 𝑇 (𝐿) = 𝐿.
For the transfer equation (2.12), I search for the asymptotic solution 𝐹
in the following form:𝐹˜ (𝑧) = 𝐿 + 𝜀 + 𝑎2 𝜀2 + 𝑎2 𝜀3 + ..

(6.2)

𝜀 = exp(𝑘𝑧)

(6.3)

where
Here, 𝑘 is constant, that bas sense of the increment, and 𝑎 refer to someset of real constants, that do not depend on 𝑧 . As it is indicated in thePreamble of this chapter, I keep in mind the case when 𝑇 (𝑧 * ) = 𝑇 (𝑧)*и 𝐹 (𝑧 * ) = 𝐹 (𝑧)* ; however, in principle, the expansion (6.3) can be used
for more complicated case too (and in this case the coefficients 𝑎 should
be complex).
The consideration could be generalised, adding new terms in to the righthand side equation (6.2), for example, terms, that are not polynomialwith respect to 𝜀. Following my wishes about the popularity 3 , below I
consider the simplest (but still non-trivial) case.
Substituting 𝐹 → 𝐹˜ into the transfer equation (2.12), which is

𝐹 (𝑧 + 1) = 𝑇 (𝐹 (𝑧))in the left hand side I get the following𝐹˜ (𝑧 +1) = 𝐿 + e𝑘 𝜀 + 𝑎2 e2𝑘 𝜀2 + 𝑎3 e3𝑘 𝜀3 + ..

(6.4)

and the right hand side gives

𝑇 (𝐹˜ (𝑧)) = 𝐿 + 𝑇 ′ · 𝜀 + 𝑇 ′ · 𝑎2 𝜀2 + 𝑇 ′ · 𝑎3 𝜀3 + ..
𝑇 ′′
𝑇 ′′′
2
2
+ (𝜀+𝑎2 𝜀 +..) +(𝜀+..)3 + ..
(6.5)
2
6
where 𝑇 ′ = 𝑇 ′ (𝐿), 𝑇 ′′ = 𝑇 ′′ (𝐿), 𝑇 ′′′ = 𝑇 ′′′ (𝐿), .. are derivatives of thetransfer function 𝑇 at the fixed point 𝐿.
coefficients at the same power of 𝜀 in expression (6.4) and in expression(6.5) should be equal. This gives the equations for increment 𝑘 andcoefficients 𝑎:e𝑘 = 𝑇 ′e2𝑘 𝑎2 = 𝑇 ′ 𝑎2 + 𝑇 ′′ /2
e3𝑘 𝑎3 = 𝑇 ′ 𝑎3 + 𝑇 ′′ 𝑎2 + 𝑇 ′′′ /6..
3 Here, the term “popularity” may have each of the two its meanings
62

(6.6)
(6.7)
(6.8)

The chain of these equations determines𝑘 = ln(𝑇 ′ )𝑇 ′′ /2
𝑎2 =(𝑇 ′ − 1)𝑇 ′𝑇 ′′ 𝑎2 + 𝑇 ′′′ /6
𝑎3 =((𝑇 ′ )2 − 1)𝑇 ′
..

(6.9)
(6.10)(6.11)

Typically, for a simple special function 𝑇 , the Mathematica, the Maple
or any similar software allow to evaluate tens of coefficients 𝑎 in realtime.4 The truncated series in representation (6.2) provides good approcimation for 𝐹 at 𝜀 ≪ 1. For positive values 𝑘 , this corresponds tothe large negative values of ℜ(𝑧). For other values, the approximation
can be improved with

𝐹 (𝑧) ≈ 𝑇

𝑛

(︁

)︁
˜
𝐹 (𝑧 − 𝑛)(6.12)

for the large enough positive values of 𝑛 at positive 𝑘 , and for the large
enough negative values of 𝑛 at negative 𝑘 . Case 𝑘 = 0 is qualified
as exotic and is considered below in the special chapter about exotic
iterates.
In may cases, representation (6.12) allows to evaluate the superfunction with the required precision. In particular, this allows to evaluate
the super exponentials to base 𝑏 < exp(1/e) [61], superfactorial [66],and holomorphic extension of the logistic sequence [70], and even theholomorphic extension of the Collatz subsequence [112]. Some of theseexamples are considered in the following chapters.
I call this (︀case “regular)︀ iteration”, as the iteration by (2.14), id est,𝑛−1
𝑇 (𝑧) = 𝐹 𝑛+𝐹 (𝑧) is holomorphic (“regular”) function in vicinity ofthe stationary point 𝑧 = 𝐿. For real fixed point 𝐿, I expect, namely regular iteration corresponds to the physically-meaningful solution 𝐹 of thetransfer equation (2.12). This statement can be qualified as conjecture.
Following the TORI axioms, I tried to negate, refute this conjecture,but I got confirmations instead. These confirmations, examples form
the significant part of this Book.
4 My experience indicates, that, for applications, the evaluation of all quantities in real time isso important, as the real holomorphism of the functions. In such a way, I try to be realistic, dealing
with real quantities in the real time. However, even for the real-holomorphic superfunctions, their
behaviour at complex values of argument is important and should be considered; but even in thiscase I try to keep algorithms short and fast, to make evaluations in the real time.

63

2

Exact solution

The approximate equality in expression (6.12) should not make an impression, that the regular iteration gives only approximation of the superfunction.5 For the case of ℜ(𝑘) > 0 (or, in particular, for 𝑘 > 0), theexact superfunction can be expressed with limit(︀)︀
𝐹 (𝑧) = lim 𝑇˜𝑛 𝐹𝑚 (𝑧 − 𝑛)𝑛→∞

(6.13)

where

𝐹𝑚 = 𝐿 + 𝜀 +𝑚∑︁𝑎𝑛 𝜀 𝑛(6.14)ℓ=2

and 𝑘 , 𝜀 and coefficients 𝑎 are defined in the previous section; while𝑚 ≥ 2 is some integer number. Due to the asymptotic properties ofthe solution 𝐹˜ , the limit in expression (6.13) does not depend on 𝑚.
However, the convergence is faster for larger 𝑚. Similar limit can be
written for the case ℜ(𝑘) < 0.
The series (6.3) can be inverted, giving the asymptotic approximation
˜ of the abelfunction 𝐺 = 𝐹 −1 . Then, the exact abelfunction can be𝐺
expressed with

(︂

)︂
1˜
𝐺(𝑧) = lim ln
𝐺𝑚 (𝑇 −𝑛 (𝑧)) + 𝑛𝑛→∞𝑘(6.15)

˜ 𝑚 is some truncation of the asymptotic expansion for the abelwhere 𝐺
function. The iterate of the transfer function, 𝑇 𝑛 (𝑧) = 𝐹 (𝑛 + 𝐺(𝑧))happen to be regular in vicinity of 𝑧 = 𝐿. In the following chapters,
this statement is verified for many specific realisations of the transfer
function 𝑇 .
Before to apply the regular iteration to the new transfer functions, forwhich the superfunctions cannot be easy expressed through the superfunctions, known in century 20, it worth to check the method for someeasy function, for which the answer is known. First, I use the method of
regular iteration to construct the superfunction for the transfer function
Doya, considered in the previous chapter. I do it, as if I would not knowthat its superfunction is Tania. This regular iteration is considered in
the next section, and compared to the exact solution Tania.
5 One of my coauthors, until now, believes, that even 𝜋 is approximate number. This can beconsidered as a kind of a mental illness, as well as some kind of religion.. However, if for any given

𝛼 > 0, the error of evaluation of some quantity can be done smaller, than 𝛼, then the quantity is
considered as exact. Then, we may consider the CPU time and number of flops required to getsome given precision of the approximate evaluation of superfunction 𝐹 in (6.12).
64https://mizugadro.mydns.jp/t/index.php/File:Doya1T.jpg

In this section, I show, how does the reg- 𝑦ular iteration work. I use the example
with transfer function Doya, describedin the previous chapter. For real valueof argument, graphic of this function is2
shown in figure 6.1 with thick curve.

𝑥)

Example with known solution: again Doya𝑦=𝑦 = e𝑥Doya(

3

For transfer function 𝑇 = Doya, thesuperfunction is known, it is function
𝐹 = Tania, and the corresponding abel- 1function is 𝐺 = ArcTania. Properties ofthese functions are described in publications [86, 108, 109] and in the previous
𝑦 = e 𝑧 − e(e−1)𝑥2
chapter. But, in order to check the reg1
𝑥ular iteration, assume for a minute, that 0
Fig. 6.1: Function Doya andwe do not know the analytic expression
its polynomial approximations
for the superfunction, and want to construct it with the regular iterations described above. After to constructit, we may compare the result with the special function Tania.
Let the transfer function, at least in some vicinity of the real values of
the argument, can be expressed with formula (5.12), I repeat it here:

𝑇 (𝑧) = Doya(𝑧) = LambertW(1+𝑧 e𝑧 )(6.16)

Properties of function LambertW are known [113, 114, 115, 116]. Thisfunction can be used even without to refer to function Tania; this emulates situation, when the superfunction 𝐹 is not known.
Graphic of function Doya is shown in figure 6.1 with thick curve. Thistransfer function describes some idealised amplifier with saturation, neglecting the spontaneous emission (and many other physical effects). In
this case, the fixed point 𝐿 = 0 should be considered; 𝑇 (0) = 0. At
small intensity at the input, the amplification coefficient is e, but it saturates at the intensity of order of unity. For small argument, the transfer
function can be expanded as follows:

𝑇 (𝑧) = Doya(𝑧) = e 𝑧 − e (e−1)𝑧 2 + 𝑂(𝑧 2 )(6.17)In this case, 𝑇 ′ (0) = e and 𝑇 ′′ (0) = −2e(e−1). The corresponding linear
and quadratic approximations are shown in figure 6.1 with thin lines.
65

2))
𝑇3
(ex
p(𝑥
𝑇4
(ex − 3))
p(
𝑥−
4))

exp
(𝑥 −
𝑇 2(

1))

2

𝐹

𝑇 (e

exp
(𝑥)

xp(
𝑥−

3

)
(𝑥

𝑇 4 (exp(𝑥−4)−exp(2(𝑥−4)))𝑇 3 (exp(𝑥−3) − exp(2(𝑥−3)))

10

𝑇 2 (exp(𝑥−2) − exp(2(𝑥−2)))
𝑇 (exp(𝑥−1) − exp(2(𝑥−1)))exp(𝑥) − exp(2𝑥))

−2

−1

0

1

2

3

4

𝑥

https://mizugadro.mydns.jp/t/index.php/File:SimudoyaTb.png
˜ by (6.12) for 𝑛 = 0..4
Fig. 6.2: Superfunction 𝐹 = Tania and its approximations 𝐹

For regular iteration of transfer function Doya, using formulas (6.9) and(6.10), I found 𝑘 = 1 and 𝑎2 = −1. In such a way, the primary approximation with singe term is𝐹˜ (𝑧) = exp(𝑧)(6.18)and the primary approximation with two terms is𝐹˜ (𝑧) = exp(𝑧) − exp(2𝑧)(6.19)

These two primary approximations are shown in figure 6.2 with thehighest and lowest curves. At the same figure, the four iterations ofthese primary approximations by (6.12) are shown for 𝑛 = 0..4. These
approximations approach the exact solution

𝐹 (𝑧) = Tania(𝑧 −1) = WrightOmega(𝑧)(6.20)

This function 𝐹 is shown in figure 6.2 with thick line. It remains between
approximations obtained with iterate from the primary approximation
𝐹˜ with single term, id est, (6.18) and those with tho terms by (6.19).
Example with the transfer function Doya shows the efficiency of theregular iteration. In the following chapters, the regular iteration is usedalso for other transfer functions; I mean, for the cases, when the superfunction cannot be easy expressed in terms of the special functions,known in century 20.
66

4Schröder equation

For the regular iteration, instead of superfunctions (presented in this Book), another formalism can be used, namely,
the formalism of the Schroeder (Schröder) functions; theyFig. 6.3
are called after Ernest Schröder, shown in figure 6.3, andE.Schröderdescribed many years ago [11, 27].
Let transfer function 𝑇 be real-holomorphic, and let 𝐿 = 0 be its fixed
point. The generalization to the case of other values of 𝐿 is straightforward, so, here I consider only case 𝐿 = 0. Let 𝑆 = 𝑇 ′ (0). The
schröderfunction is solution 𝑔 of the Schröder equation𝑔(𝑇 (𝑧)) = 𝑆 𝑔(𝑧)(6.21)

where 𝑠 is some constant. Usually, it is assumed that 𝑇 (0) = 0, id
est, zero is fixed point of the transfer function 𝑇 . One can search forsolutions in the asymptotic form (6.21)𝑔(𝑧) =∞∑︁

𝑎𝑝 𝑧 𝑝(6.22)

𝑝=1

where coefficients 𝑎 are constants, id est, do not depend on 𝑧 . Usually,
the series in the right hand side of (6.22) diverges, but this representation can be used for approximation of 𝑔 at small values of |𝑧|. Then,any required precision can be achieved, applying the Schroeder equation(6.21) or its inverse(︁𝑔(𝑧) = 𝑆 𝑔 𝑇

−1

)︁
(𝑧)

(6.23)

looking, what is smaller, |𝑇 (𝑧)| or |𝑇 −1 (𝑧)|. This method is especially
explicit for a real-holomorphic transfer function 𝑇 , that monotonously
grows along the real axis.
In certain parti f the complex plane, the abelfunction 𝐺 can be expiressed through the Schroeder function 𝑔 :𝐺(𝑧) = log𝑆 (𝑔(𝑧))(6.24)Taking logarithm to base 𝑆 of the both sides of equation (6.21), I get

𝐺(𝑇 (𝑧)) = 1 + 𝐺(𝑧)(6.25)

this equation is just Abel equation for the same transfer function 𝑇 .
In the similar way one can express the analogy of superfunction, it canbe called “scaling function” or even “scalingfunction”. As examples of the
67

5
4

𝑢=

𝑣 =1

𝑦
2

3
2 𝑣
1

=2

0

cut

−1

2
−
=

−2 𝑣

𝑢=

1

𝑣 =0

−3

𝑣 =−

−4
−5

1

−6
−6 −5 −4 −3 −2 −1 0

1

2

3

https://mizugadro.mydns.jp/t/index.php/File:Olga6map.jpg
Fig. 6.4:

4

5 𝑥

𝑢+i𝑣 = Olga(𝑥+i𝑦) by (6.26)

Schroeder functions and the scaling functions, figure 6.4 and 6.5 showcomplex maps of functions Olga and Anka. They are Schroeder function
and the scaling function for the transfer function 𝑇 = Doya, consideredin chapter 5 and in the previous section.
The Schroeder function Olga can be expressed through function Tania:(︁)︁Olga(𝑧) = Tania ln(𝑧)(6.26)The Scaling function Anka can be expressed through the ArcTania:(︁)︁
Anka(𝑧) = exp ArcTania(𝑧)(6.27)For the transfer function Doya, the scaling factor 𝑆 = e; so, the natural logarithm and the natural exponent appear in equations (6.26) and(6.27). Definitions (6.26) and (6.27) imply, that olga = anka−1 .
68

𝑦

𝑢=0

5
4

𝑣 =0

3

𝑣 =0
𝑢=0

𝑣 = 16𝑣 = −16

𝑣=

0

0

1

𝑣 =0

𝑢=

0

1

𝑢=0
𝑢=

2

𝑣 = 16𝑢 = − 16𝑢 = −16
𝑢 = 16𝑣 = 16𝑣 = −16
𝑢 = 16𝑢 = −16𝑣 = −16𝑣 = 16𝑢 = −16
𝑢 = 16

−1
−2

𝑢=0

−3
−4
−5
−6
−6 −5 −4 −3 −2 −1

0

1

2

3

4

https://mizugadro.mydns.jp/t/index.php/File:Anka616map.jpg
Fig. 6.5:5𝑥

𝑢+i𝑣 = Anka(𝑥+i𝑦) by (6.27)

In this section, I use the example, when some dependences can be expressed in terms of elementary functions. In particular, this refers tofunction ArcTania by equation (5.4); in the most of the complex plane(except the negative part of the real axis), ArcTania(𝑧) = 𝑧 + ln(𝑧) − 1.
Then, function Anka can be expressed as follows

Anka(𝑧) =𝑧
exp(𝑧)e(6.28)

Functions Anka and Olga satisfy the scaling equation

Doya(Anka(𝑧)) = Anka(e 𝑧)(6.29)and the Schroeder equation

Olga(Doya(𝑧)) = e Olga(𝑧)69(6.30)

The Reader is invited to investigate the ranges of validity of equations

Olga(Anka(𝑧)) = 𝑧
Anka(Olga(𝑧)) = 𝑧(6.31)(6.32)

In wide range, that includes the positive part of the real axis, iterates
of function 𝑇 = Doya can be expressed through its scaling function
𝑓 = Anka and Schröder function 𝑔 = Olga = Anka−1 = ArcAnka :𝑇 𝑛 = 𝑓 (𝑆 𝑛 𝑔(𝑧))(6.33)

for some appropriate constant 𝑆 . Readers are invited to check, that
formula (6.33) gives the same iterates, as the expression through thesuperfunction 𝐹 = Tania and Abel function 𝐺 = ArcTania

𝑇 𝑛 = 𝐹 (𝑛 + 𝐺(𝑧))(6.34)

While the real-holomorphic transfer function 𝑇 has real fixed point 𝐿with the scaling factor 𝑆 , and this fixed point is used to construct thesuperfunction 𝐹 and the scalingfunction 𝑓 , formulas (6.33) and (6.34)
are equivalent. Several examples from this book, considered with superfunctions and abelfunctions, can be treated also with scalingfunctions
and schroöderfunctions.
Relation (6.34) is more general than (6.33). Here I announce few cases,
when the Schroeder functions fail.
If 𝑇 ′ (𝐿) = 1, then the scaling factor 𝑆 becomes unity, and expression(6.24) fails. In particular, this is case of 𝑇 (𝑧) = exp(𝑧/e), case of 𝑇 (𝑧) =zex(𝑧) = 𝑧 exp(𝑧), and that for 𝑇 (𝑧) = sin(𝑧), see Table 3.1.
The transfer function 𝑇 may have no real fixed points, as it takes place
for 𝑇 = exp.
In addition, it may happen, that the transfer function 𝑇 has no fixed
points at all, as it takes place for 𝑇 (𝑧) = tra(𝑧) = 𝑧 +exp(𝑧).
Iterates of these functions 𝑇 are straightforward with the superfunctions
and abelfunctions. These examples are mentioned in the Table 3.1 andconsidered in the following chapters of this Book. However, these cases
are difficult to treat with the scalingfunctions and the schroederfunctions, if at al. The Schroeder functions, if they can be applied, do notgive any new in compare to use of the superfunctions. For this reason,
this Book is dedicated to superfunctions and abelfunctions, and not toschroederfunctions.
I hope, the example above is sufficient to feel relation between Abelfunctions and Schroederfunctions. On this point I stop speculations about
the Schroeder functions, Schroeder equations, Scaling equations, andscaling functions, and return to superfunctions. Superfunctions for the
specific quadratic transfer function are considered in the next chapter.
70

Chapter 7
Logistic map
Term “Logistic map” 1 may refer to the quadratic transfer
function

𝑇 (𝑧) = Elu𝑠 (𝑧) = 𝑠 𝑧 (1−𝑧)

(7.1)

Fig.

7.1:

P.V.Elutin

Usually, parameter 𝑠 is assumed to be a real number. Term“map” is used also in the context of the “complex map”, and this may
cause confusions. For this reason I give this function name Elu𝑠 . The
thee characters of the name are taken from the last name of Pawel
Elutin (see Fig. 7.1), my teacher of Quantum Mechanics. He asked me
to construct the analytic extension of iterates of function 𝑇 by (7.1) inthe private communication [58]. In such a way, this chapter could becalled also “Elutin function”. Here I present some results of publication[70] that appeared as the answer on the request by Elutin.

Historically, iterates of another transfer function had been considered
before the iterates of the Elutin function. I mean, iterates of factorial[66]. The request to do the same of the logistic map appeared as result of
the publication about the half iterate of factorial. In this Book, I do notfollow the history; first, I describe the superfunction, abelfunction anditerates of the logistic map, or iterates of the Elutin function by (7.1);
these iterates seem to be simpler than those for the factorial. For various
values of parameter 𝑠, iterates of Elu𝑠 are shown in figure 7.2. In orderto plot this figure, I use the specific superfuction and the corresponding
abelfunction. I describe them in this chapter.
The logistic sequence is solution 𝐹 of the logistic equation

𝐹 (𝑧 +1) = Elu𝑠 (𝐹 (𝑧))1 https://en.wikipedia.org/wiki/Logistic_map

71

(7.2)

1

1

1

1
2

0

1
2

𝑠=3
0

0.5

1

0

1
2

𝑠=4
0

0.5

𝑛=
1
𝑛=
𝑛
0
.
𝑛= =
0.5 8
0.2

𝑦

𝑛
𝑛= = 1𝑛𝑛 = = 0 0.8.
0.2 5

𝑦 = 𝑇 𝑛 (𝑥)𝑛 =𝑛 =𝑛𝑛 = = 0. 1
0 8
0. .5
2

𝑦 = 𝑇 𝑛 (𝑥)

1

0

𝑠=5
0

0.5

1

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi1a345T300.png
𝑦 = Elu𝑛𝑠 (𝑥) for 𝑠 = 3, left; for 𝑠 = 4,center; for 𝑠 = 5, right; curves for 𝑛 = 1, 𝑛 = 0.8, 𝑛 = 0.5, 𝑛 = 0.2 are drawn.

Fig. 7.2: Iterates of the Elutin function (7.1):

Equation (7.2) is, actually, the transfer equation (2.12), with the Elutinfunction (logistic map) Elu𝑠 by (7.1) as the transfer function 𝑇 . In orderto define the sequence, the initial value 𝐹 (0) should be specified.
In publications about the logistic equation (7.2) with transfer function(7.1), the argument of function 𝐹 is assumed to be an integer number[25, 29, 31, 56]. For integer argument of the solution 𝐹 of equation (7.2),
the solution catches some properties of transition of the physical systems
to chaos [55, 22, 21]. Iterates of the transfer function 𝑇 by (7.1) appear
as a rough description of the stochastic physical systems, as a simple,
heuristic approach for the problems of hydro- and aero- dynamics, andalso the transition regime of the stochastic lasers, in vicinity of the singlemode regime of generation. Here I consider the holomorphic extensionof the logistic sequence for not only integer, but complex values of theargument.

1

Logistic sequence

Iterates of the Elutin function (logistic map) 𝑇 = Elu𝑠 by (7.1), asfunctions of real argument, are shown in figure 7.2 for 𝑠 = 3 (left picture)𝑠 = 4 (central picture) and 𝑠 = 5 (right picture); 𝑦 = Elu𝑠𝑛 (𝑥) is plottedversus 𝑥 for 𝑛 = 0.2, 𝑛 = 0.5, 𝑛 = 0.8 and 𝑛 = 1. All the graphics in figure
7.2 are plotted with the same formulaElu𝑛𝑠 (𝑧) = 𝐹 (𝑛 + 𝐺(𝑧))
72

(7.3)

1

𝐹 (𝑥)

𝑠=3
𝑠 = 3.4
𝑠 = 3.8

0

−2

−1

0

1

2

3

4

𝑥

2

3

4

𝑥

1

𝐹 (𝑥)
𝑠 = 3.90𝑠 = 4.1

−1

−2

−1

0

1

https://mizugadro.mydns.jp/t/index.php/File:LogisticSecK2.jpg

Pav𝑠 (𝑥) by formulas (7.4),(6.12), (7.8) for 𝑠 = 3, 𝑠 = 3.4, 𝑠 = 3.8,
at the top picture, and for 𝑠 = 3.9, 𝑠 = 4, 𝑠 = 4.1, bottom.

Fig. 7.3:

where 𝐹 is specific superfunction, solution of equation (7.2), and 𝐺 =𝐹 −1 is the inverse function (id est, abelfunction of Elu𝑠 ). Superfunction
𝐹 appears as holomorphic extension of the logistic sequence [70]. Superfunction 𝐹 can be constructed with the regular iteration, describedin the previous chapter. This construction is described below.
At the construction of a superfunction, the key question is about the
fixed points of the transfer function. For transfer function 𝑇 = Elu by(7.1), equation Elu𝑠 (𝑧) = 𝑧 has two solutions, 𝑧 = 0 and 𝑧 = 1 − 1/𝑠.
First of these solutions does not depend on 𝑠. This solution is used forconstruction and evaluation of the “holomorphic extension of the logistic
sequence”, id est, the specific solution 𝐹 = Pav𝑠 of equation 7.2, shownin figure 7.3 below. Then I can construct the inverse function 𝐺 = 𝐹 −1 ,which is the corresponding abelfunction. These 𝐹 and 𝐺 allow to plot
figure 7.2 by formula (7.3). Construction of these functions is describedin the next section.

2

Fixed point 𝐿 = 0

For the Elutin transfer function by (7.1), with the regular iteration at
the fixed point 𝐿 = 0, I construct the superfunction 𝐹 = Pav𝑠 . Graphics
𝑦 = Pav𝑠 (𝑥) are shown in figures 7.3 for different 𝑠. The complex maps
are shown in figures 7.4, 7.5, 7.6. Below I describe the construction andevaluation of this superfunction.
73

𝑦 𝑣 =0
2
𝑢=0

𝑣 =0

𝑣 = −0.1𝑣 =0𝑣 = 0.1

𝑣 =0

𝑢 = 0.2

0 𝑣 =0

𝑢 = 0.1

1

−1
𝑢=0
−2
𝑣 =0
−2

−1

0

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2c3T1000.png
Fig. 7.4:

𝑢+i𝑣 = Pav3 (𝑥+i𝑦) by (7.8)

I need some name to denote superfunction for the transfer function Elu𝑧 ;For this superfunction, I suggest name “Pav”. In this name, I use first
2.5 characters of the first name Pawel of my teacher (who had asked me
to construct this function; half of letter w appears as v). First threecharacters of his last name Elutin are already used below to denote thequadratic function (7.1), known also as “logistic map”. My excuse fordefining of the new name is the following: the “logistic map”, as it iscalled in the literature, is not actually map in the common sense of thisword; it is holomorphic function; and the similar note refers to the socalled “logistic sequence”. So, I use term “Elutin function” instead of“logistic map”; in the context of this book, it is holomorhic function,
just some polymomial.

74

𝑦
2

𝑣 =0

0 𝑣 =0𝑣 =0𝑣 = 0 .1
𝑣 = −0.1

𝑣 =0

𝑢 = 0.1
𝑢 = 0.2

1 𝑢=0−1 𝑢 = 0
−2

𝑣 =0
−2

−1

0

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2c4T1000.png
Fig. 7.5:

𝑢+i𝑣 = Pav4 (𝑥+i𝑦) by (7.8)

In order to define function Pav𝑠 , first, I construct its asymptotic. For
the fixed point 𝐿 = 0, in the expansion (6.2), increment 𝑘 = log 𝑠, andthe expansion parameter 𝜀 = 𝑠𝑧 ; This gives the primary expansion of thesuperfunction𝐹˜ (𝑧) =𝑁−1∑︁

𝑎𝑛 𝑠𝑛𝑧 + 𝒪(𝑠𝑁 𝑧 )

(7.4)

𝑛=1

For the representation (6.2), I set also 𝑎1 = 1. Variation of this parameter causes only the displacement of the argument of the resulting
superfunction; this does not affect the iterates of the transfer function75𝑦
2 𝑣 =00 𝑣 =0𝑢=1
𝑣 =0

𝑢 = 0.1
𝑢 = 0.2

1 𝑢=0−1 𝑢 = 0−2 𝑣 = 0

−2

−1

0

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2c5T1000.jpg
Fig. 7.6:

𝑢+i𝑣 = Pav5 (𝑥+i𝑦) by (7.8)

Elu𝑠 . Expressions (6.10) determine the coefficients 𝑎. In particular,
𝑎2 =

1
1−𝑠

(7.5)

2
(1−𝑠)(1−𝑠2 )5+𝑠=(1−𝑠)(1−𝑠2 )(1 − 𝑠3 )

𝑎3 =

(7.6)

𝑎4

(7.7)

The primary representation 𝐹˜ (𝑧) by (7.4) allows the accurate (precise)evaluation of 𝐹 (𝑧) at large negative values of ℜ(𝑧). Then, the continual
extension of the logistic sequence, shown in figure 7.3, appears as limit

𝐹 (𝑧) = Pav𝑠 (𝑧) = lim Elu𝑠𝑛
𝑛→∞
76

(︁

)︁
˜
𝐹 (𝑧 −𝑛)(7.8)

For real argument, graphics of superfunction 𝐹 = Pav𝑠 are shown infigure 7.3 for 𝑠 = 3, 𝑠 = 3.4, 𝑠 = 3.8 at the top picture and for 𝑠 = 3.9,
𝑠 = 4, 𝑠 = 4.1 at the bottom picture. For 𝑠 = 3, 𝑠 = 4 and 𝑠 = 5, thecomplex maps of function 𝐹 = Pav𝑠 are shown in figures 7.4, 7.5, 7.6
announced above.
While 𝑠 < 3.5 (id est, does not exceed the “Pomequ-Mannevill constants” [70, 19, 53] ), the logistic sequence 𝐹 (as for integer values of theargument, as for the real values) shows pretty boring and regular oscillations. At larger values of 𝑠, at the argument grows, the oscillations
become dense. Observation of 𝐹 (𝑛) = Pav𝑠 (𝑛) at the integer values of𝑛 make an impression of quasi-random sequence. While 𝑠 ≤ 4, at real𝑥, function 𝐹 (𝑥) oscillates within segment [0,1], and only at 𝑠 = 4 ittouches the borders of this segment. At 𝑠 > 4, the function has “gaps”;
they become deeper and deeper at the increase of the argument 𝑥 orparameter 𝑠.
At 𝑠 = 4, the holomorphic extensiion 𝐹𝑠 = Pav𝑠 of the logistic sequencecan be represented as the elementary function,

Pav4 (𝑧) = (1 − cos(2𝑧 ))/2

(7.9)

The curve, corresponding to 𝑠 = 4 at figure 7.3, could be plotted even
without the regular iteration; the same applies to map at figure 7.5.
At consideration of the holomorphic extension of the logistic sequencein the complex plane, its behaviour is regular. However, at large 𝑠,
the oscillations become more and more dense with the increase of theargument.
Superfunction 𝐹𝑠 , built with regular iteration at the fixed point 𝐿 = 0,
is entire and periodic. The period is pure imaginary (for real 𝑠):

𝑃 = 2𝜋i/ ln(s)(7.10)

This periodicity is seen in figures 7.4, 7.5, 7.6: The isolines are reproduced at the corresponding translations along the imaginary axis. A
little bit more than one period fit the height of the map in figures 7.4,
7.5, 7.6.
With holomorphic extension of the logistic sequence, id est, superfunction Pav𝑠 , one can construct the non-integer iterates of the logistic
map, id est, iterate the Elutin function Elu𝑠 by (7.1); in particular,
the half iterate of this transfer function can be constructed. However,for this, the inverse function is also required, I mean, the abelfunction
𝐺𝑠 = ArcPav𝑠 = Pav−1
𝑠 .
77

𝑣 =2

𝑦
2

𝑢=

1

0

𝑣 =1

cut

0

𝑢=

−1

𝑣 = −1

−1

2
𝑣 =−

0

−2

−2

cut

0

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2d3t1500.jpg
Fig. 7.7:

𝑢+i𝑣 = ArcPav3 (𝑥+i𝑦)

The complex maps of the abelfunction ArcPav𝑠 are shown in figures 7.7,
7.8, 7.9 for 𝑠 = 3, 𝑠 = 4 and 𝑠 = 3. This abelfunction is described in thenext section.

3

Abelfunction for the Elutin function

This section describes the Abel function of the Elutin function Elu𝑠 by(7.1). I call this function ArcPav𝑠 . It is inverse function of function
Pav𝑠 by (7.8), described in the previous section.
Function 𝐺 = 𝐺𝑠 = ArvPav𝑠 = Pav−1
𝑠 satisfies the Abel equation

𝐺𝑠 (Elu𝑠 (𝑧)) = 𝐺𝑠 (𝑧) + 1
78(7.11)

𝑦
2
1

𝑣=

𝑢=0

2

𝑣 =1

0 cut
−1

=
𝑣
−2

cut

2
−

−2

𝑣 = −1

−1

0

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2d4t1500.jpg
Fig. 7.8:

𝑢+i𝑣 = ArcPav4 (𝑥+i𝑦).

This equation is the same as (2.13), the only additional subscript 𝑠
is gaffed to indicate the parameter in the transfer function 𝑇 = Pav𝑠 .
Complex maps of function 𝐺𝑠 are shown in figures 7.7, 7.8, 7.9 for various
values of 𝑠. This section describes evaluation of this function.

˜ for abelfunction 𝐺 can be found with inThe asymptotic expansion 𝐺
version of expansion (7.4) for superfunction 𝐹 ; it has the following form:˜𝐺(𝑧)= log𝑠(︃𝑁 −1∑︁)︃
𝐶𝑛 𝑧 𝑛 + 𝑂(𝑧 𝑁 )(7.12)

𝑛=1

In the software Mathematica, there is special procedure InverseSeries
for such inversions.
Coefficients 𝐶 in equation (7.12) depend on parameter 𝑠, but this de79

=
𝑣

2
1

𝑢

1

0
=

0 cut

cut

−1

𝑣=

−2

−2

−1
Fig. 7.9:

0

−1
1

2

https://mizugadro.mydns.jp/t/index.php/File:Logi2d5t1500.jpg

𝑦𝑥

𝑢+i𝑣 = ArcPav5 (𝑥+i𝑦).

pendence is not indicated explicitly, in order to keep the expression
compact. The same coefficients can be found also substituting the expansion (7.12) into the Abel equation (7.11) with transfer function Elu𝑠and equalizing the coefficients at equal powers of 𝑧 in the left and in theright hand sides. The resulting coefficients 𝐶 appear as follows(7.13)

𝐶1 = 1
𝐶2 =1
𝑠−1(7.14)

3𝑠(𝑠 − 1)(𝑠2 − 1)(𝑠2 −5)𝑠=(𝑠−1)(𝑠2 −1)(𝑠3 −1)

𝐶3 =(7.15)

𝐶4(7.16)...

80

The truncated series of expansion (7.12) gives way to evaluate abefunction 𝐺 at small values of the argument. At large values, the representation can be extended with−𝑛˜𝐺(𝑧) ≈ 𝐺(Elu𝑠 (𝑧)) + 𝑛(7.17)

for some large enough integer 𝑛. The negative integer iterates of thetransfer function can be evaluated, using representation of the inverseof the transfer function:

1
ArcElu−1−𝑠 (𝑧) = ArcElu𝑠 (𝑧) =
2

√︂

1 𝑧
−
4 𝑠

(7.18)

Symbol “ ≈” in expression (7.17) should not make an impression, that
only the approximation of the Abel function is constructed. The exact
Abel function can be expressed through the limit

(︁

)︁
−𝑛
˜
ArcPav𝑠 (𝑧) = 𝐺(𝑧) = lim 𝐺(Elu𝑠 (𝑧)) + 𝑛𝑛→∞(7.19)

˜ 𝑠 denotes the truncated series in the expansion (7.12).
where symbol 𝐺
This representation is used in generators of figures 7.7, 7.8, 7.9 . The
same representation is used to plot iterates of the transfer function infigure 7.2. at 𝑠 = 3, 𝑠 = 4 and 𝑠 = 5.
Function ArcElu𝑠 (𝑧) by (7.18) has the branch point 𝑧 = 𝑠/4. This
determines also the branch of the abelfunction ArcPav𝑠 and cuts in theright hand sides of maps in figures 7.7, 7.8, 7.9 .
The non-integer iterates of the transfer function, shown in figure 7.2,
have the similar branching. The half iterate is special case of the noninteger iterate; this case shown in figures 7.10, 7.11, 7.12 and describedin the next section.

4

Halfiterate of logistic operator

With superfunction 𝐹𝑠 and abelfunction 𝐺𝑠 = 𝐹𝑠−1 , iterates of the transfer function 𝑇𝑠 are expressed with formula (7.3). This representation isused to generate figure 7.2. In particular, 𝑇𝑠0.5 by (7.3) is solution of the
problem, formulated by P.Elutin: The function ℎ𝑠 is built asℎ𝑠 = Elu0.5𝑠(7.20)

such that its second iterate gives the logistic operator (7.1); id est, forsome range of values of 𝑧 , the relation below holds:ℎ2 (𝑧) = ℎ(ℎ(𝑧)) = 𝑇 (𝑧) = Elu𝑠 (𝑧) = 𝑠 𝑧 (1−𝑧)81(7.21)

𝑣=

0

𝑦
2

=4
𝑣
1
0 𝑣 =0
−1

−1

0

−2

𝑣=

−2

𝑢=−
4
𝑢=
−2
𝑢=
0
𝑢=
2

𝑣 = −40

1

2𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2b3t1000.jpg
Fig. 7.10:

1/2

𝑢+i𝑣 = Elu3 (𝑥+i𝑦).

Complex maps of function ℎ𝑠 are shown in figures 7.10, 7.11 and 7.12
for various values of parameter 𝑠. These maps look similar, because, atlarge values of the argument 𝑧 , the approximate relation takes place:Elu𝑠 (𝑧) ≈ 𝜏 (𝑧) = −𝑠(−𝑧)2(7.22)

Iterates of function 𝜏 by (7.22) can be calculated in analogy with iteratesof the transfer function by (4.19).
The readers are invited to plot the superfunction and the abelfunction
for the transfer function 𝜏 . This can be done in analogy with the iteratesof the power function, presented in Table 3.1. In particular, the half
iterate can be written as follows:

𝜏

1/2

√

(𝑧) = −𝛼(−𝑧) 2
82(7.23)

𝑣=

0𝑦2
1 𝑣 =4

𝑣 =0

−1𝑣

= −4

−1

0

0

−2

𝑣=

−2

𝑢=−
4
𝑢=
−2
𝑢=
0
𝑢=
2

0

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2s4t.jpg
Fig. 7.11:

𝑢+i𝑣 = Elu0.5
4 (𝑥+i𝑦)√where 𝛼 = 𝑠1/(1+ 2) is constant, and this constant depends on parameter𝑠 slowly.
Figure 7.13 shows the map of function 𝜏 1/2 by formula 7.23 at 𝛼 = 1.8;
it should be compared to maps 7.10, 7.11 and 7.12 , that correspond tovarious values of 𝑠. At large values 𝑥2+𝑦 2 , the only quadratic term in the
expansion of the transfer function is important, and all the four maps
at figures 7.13, 7.10, 7.11 and 7.12 look similar. In order to make thedifference seen, these maps occupy all the width of the page available.
When the non-integer iterate of some function is constructed, it has
sense to check the results of iteration of this iterate. For the transfer
function Elu𝑠 and its half iterate ℎ = Elu0.5
𝑠 , such a check, test is shown
83

𝑣=

0

𝑦
2
1

𝑣 =4

0 𝑣 =0

𝑢=

−1

0

4

0
𝑢=

−2

𝑣=

−2

−4

𝑢=−
4

−1 𝑣 =

0
1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2s5t.jpg
Fig. 7.12:

𝑢+i𝑣 = Elu0.5
5 (𝑥+i𝑦)in figure 7.14. This figure shows the map pf function ℎ2 for 𝑠 = 3, 𝑠 = 4
and 𝑠 = 5, id est, for the same values of parameter, that are used infigure 7.2.
In the left hand side of maps in figure 7.14, the second iterate of functionℎ coincides with the logistic mapping. The scratched lines show the limit
of applicability of relation (7.21).
Relation ℎ(ℎ(𝑧)) = 𝑇 (𝑧) is valid at least for ℜ(𝑧) < 1/2 . In such away, the holomorphic extension of the logistic sequence, and also thecorresponding non-integer iterates of the logistic operator correspond to
the intuitive expectations about these functions.
There is nothing specific in the half iterate of a function, while it is84

1 𝑣 =4

𝑣=
𝑣= 2
𝑣= 0
𝑣 = −2
−4

𝑦

𝑣 =0

cut

2

0

−3
−3

−2

−1

0

4
𝑣= 2
𝑣= 0
𝑣 = −2𝑣=−2𝑢=𝑢 = −4
−3
𝑢=
0
𝑢=
3

−1

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logi2qt.jpg√

Fig. 7.13:

𝑢+i𝑣 = −1.8(−(𝑥+i𝑦)) 2

expressed through the superfunction and the abelfunction. It is just
non-integer iterate, while number 𝑛 of iterate is 1/2. On the otherside, in the literature, the iteration half is often considered as something
magic; perhaps, because for iterate half of some transfer function, theverification is especially simple; just iterate the halfiterate twice and see
the region, where the result coincide with the original transfer function.
Figure 7.14 appears as an example of such a verification.

√

Interest to iterate half, since the half iterate of factorial ( ! ) [36, 37]
appears as a tradition. Namely iterate half is mentioned in the titlesof publications [10, 93]. So, I follow this tradition and analyse range ofvalidity of representation of a function through its iterate half.

85

𝑦

−1

1
2

𝑣 =4
𝑣 =2
0 𝑣 =0

𝑦

−1

𝑠=5

1

0

1

cut

cut

𝑥

𝑥

𝑢
𝑢 == 4
𝑢= 3
2

𝑢=

1
2

0
𝑢
𝑢 == 4
𝑢= 3
𝑢= 2
1

0
𝑢 = −1
𝑢= 2
𝑢=− 3
𝑢=−
𝑢 = −4

𝑠=4

𝑣 =0

0
𝑢 = −1𝑢=𝑢 = −2𝑢 = −3𝑢 = −41
2 𝑣 =4𝑣 =20 𝑣 =0

cu
t

𝑠=3

cu
t

𝑢=
𝑢= 3
𝑢= 2
1

𝑣 =0

𝑦

𝑢=
𝑢 = −4𝑢 = −2
0

−1

−1

2
4
=
=
𝑢 𝑢
0

cut

𝑣 =0

1

𝑣 =4
𝑣 =2
0 𝑣 =0𝑣 = −2𝑣 = −4
1
−
2

cut

1

https://mizugadro.mydns.jp/t/index.php/File:Logiha300.jpg
Fig. 7.14:

𝑢+i𝑣 = ℎ(ℎ(𝑥+i𝑦)) by (7.20) for 𝑠 = 3, 𝑠 = 4 and 𝑠 = 5

86

𝑥

0

−4

−3

−2

𝑣 =0

1

𝑣 = −0.1𝑣 =0𝑣 = 0.1

−3

−2

−1

0

1

2

𝑥

1

2

𝑥

𝑦

𝑣 =0

𝑣 =0

𝑣 =0

𝑣 =0

𝑢 = 0.8𝑣 =0
𝑢 = 0.7

𝑣 =0

𝑢 = 0.7

0

𝑣 =0

1

−1

0

−1

−4

https://mizugadro.mydns.jp/t/index.php/File:Logi5ab400.jpg

Fig. 7.15: Superfunction Paw4 (𝑥) by (7.25),(6.12): explicit plot 𝑦 = Paw4 (𝑥), top
picture, and complex map 𝑢+i𝑣 = Paw4 (𝑥+i𝑦), bottom picture, by (7.28)

5Another fixed point, 𝐿 = 1 − 1/𝑠

Holomorphic extension 𝐹𝑠 of the logistic sequence by formulas (7.4),(6.12)
is regular and periodic; the period 𝑃𝑎 by (7.10) is pure imaginary andslowly (as logarithm) depends on parameter 𝑠. This solution is notunique. In analogy with the solution, that approaches the fixed point
𝐿 = 0, one may construct other solutions, that approach the fixed point
𝐿 = 1−1/𝑠 at minus infinity. Complex map of one of such solutions isshown in figure 7.15; this superfunction is described in the is section.
The logistic operator, id est, the transfer function 𝑇𝑠 by (7.1) has two
fixed points, 𝐿 = 0 and 𝐿 = 1 − 1/𝑠. Consider the last of these fixed
points. By the general way of regular iterations, I construct the superfunction 𝑓 , that approaches 1−1/𝑠 at minus infinity, as follows:𝑁 −1)︁𝑛𝑠 − 1 ∑︁ (︁𝑧˜𝑓 (𝑧) =+𝑑𝑛 (𝑠−2) cos(𝜋𝑧 + 𝜙)
𝑠
𝑛=1
(︁
)︁𝑁
𝑧
𝑓 (𝑧) = Paw𝑠 (𝑧) = 𝐹˜ (𝑧) + 𝑂 (𝑠−2) cos(𝜋𝑧 + 𝜙)

(7.24)(7.25)

where 𝑑 are real parameters and 𝜙 is real constant. Substitution of thisexpansion into equation (2.12) gives the chain of equations for coeffi87

cients 𝑑. As in the previous cases, I set 𝑑1 = 1; then−𝑠(𝑠 − 1)(𝑠 − 2)−𝑠2=(𝑠 − 1)(𝑠 − 2)(𝑠 − 3)−(𝑠 − 7)3 𝑠3=(𝑠 − 2)(𝑠 − 3)(𝑠3 − 8𝑠2 + 22𝑠 − 21)

𝑑2 =
𝑑3
𝑑4

(7.26)The truncated series gives the accurate approximation of function 𝐹 (𝑧),while the effective parameter of expansion, id est (𝑠−2)𝑧 cos(𝜋𝑧 + 𝜙) issmall. Truncation with 4 coefficients in (7.25) provides of order of 10
significant figures while𝜋|ℑ(𝑧)|+ln(𝑠−2)ℜ(𝑧) < 4(7.27)The range of approximation can be extended with iterations (6.12); thiscan be used both for definition and algorithm of precise evaluation of
the superfunction:𝑓 (𝑧) = Paw𝑠 (𝑧) = lim Elu𝑠𝑛 (𝑓˜(𝑧 −𝑛))𝑛→∞(7.28)

I denote this function with Paw𝑠 , in order to distinguish it from Pav𝑠in the previous sections; I hope, use of generic name 𝐹 for different
superfunctions will not cause confusions. Representation (7.28) is used
to generate figure 7.15.
Superfunction Paw𝑠 by (7.28) is asymptotically–periodic; the asymptotic period

𝑃 =2𝜋ln(𝑠−2) i + 𝜋(7.29)

in the upped half plane and 𝑃 * in the lower half plane.
In contrast with superfunction Pav𝑠 , that is built up at the fixed point
zero, for superfunction Paw𝑠 , the choice of the inverse function is notstraightforward. One needs to choose, which of the oscillations should
be used to return the value of function. For this reason I do not provide
the corresponding abelfunction here. The Reader is invited to construct
it according to own preferences.
Following the 6th of the TORI axioms, about the simplicity, I consider
the superfunction by (7.4),(7.8) as “principal”, because it seems to me
simpler than that by (7.24). For the logistic operator as transfer function, the regular iteration provides the regular superfunction; however,superfunctions, constructed at different fixed points, may show pretty
different behaviour.
88𝑦Factorial is holomorphic solution ofequation

Factorial(𝑧 +1) = 𝑧 Factorial(𝑧) (8.1)

𝑡𝑧 𝑒−𝑡 d𝑡

54Factorial can be expressed with

∫︁ ∞

6

(8.2)

3

I use this notation instead of that with
with exclamation, Factorial(𝑧) = 𝑧!,
in order to simplify indication of thenumber of iterate in the upper superscript.

2

Factorial(𝑧) =01

https://mizugadro.mydns.jp/t/index.php/File:Facsma.jpg

l( 𝑥 )

Factorial

𝑦 = Factoria

Chapter 8

0 1 2 3 𝑥
Graphic 𝑦 = Factorial(𝑥) versus 𝑥 is
Fig. 8.1: 𝑦 = Factorial(𝑥)shown in figure 8.1. At 𝑥 > 2, factorial
shows fast monotonous growth. At 0 < 𝑛 < 1, the iterates Factorial𝑛should show the similar, but slower growth. These iterates are topic of
this chapter.
This chapter describes the superfunction of factorial, denoted below as
SuFac, and the Abel function, denoted as AuFac. Then, the iterates ofFactorial appear as(︀)︀
Factorial𝑛 (𝑧) = SuFac 𝑛 + AuFac(𝑧)(8.3)

Below, I construct functions SuFac and AuFac and describe their properties. In this chapter, I retell the basic concept of publication in the
Moscow University Physics Bulletin, 2010 [66].
891Physics department

Iterates of factorial and its superfunction and abelfunctions had beenreported in 2010 [66], before iterates of the logistic map (Elutin function), described in the precious chapter. In this section I explain, why I
consider this function as important.
Past century, during the USSR, my teacher of Quantum Mechanics had
asked students to give the physical sense to the operator “square rootof factorial”. That sign was well known in the USSR as symbol of the
Physics Department of the Moscow State University shown in figure 2.4,
it is mentioned in the Introduction. In analogy with other operators of
Quantum Mechanics, “square root of factorial” should be some functionℎ such that its second iteration gives factorial, ℎ2 = Factorial, id est,
(8.4)ℎ(ℎ(𝑧)) = 𝑧! = Factorial(𝑧)One could guess, that this function should be real-holomorphic, growing faster than any polynomial but slower than any exponential. Thattime it was difficult, to evaluate such a function ℎ: the formalism of
superfunctions had not yet been developed.
In principle, the Schroeder function and the scaling function could be
used to construct and evaluate the solution ℎ of equation (8.4), insteadof superfunctions. That time, for students, dealing with Quantum Mechanics, it was difficult to guess, that for the square root of factorial,
the Schroeder (Schröder) equation should be used instead of the widely
known Schroedinger (Schrödinger) one 1 . In addition, that time, no algorithms for evaluation of the scaling function and the Schroeder function
were available.

√

After the successful evalation of tetration to base 2, reported in 2009 at
the journal Mathematics of Computation [61], the problem with iterates
of factorial (and, in particular, of the half iterate) happen to be pretty
solvable. The solution is published in the Moscow University Physics
Bulletin [66] and described below.
I feel, first I should remind the properties of factorial. This is matter ofthe next section.
1 Iterates of factorial and√! can be evaluated also with the scaling function 𝑓 and the Schroeder

function 𝑔 by (6.33)
902

Factorial and its fixed points

For real argument, the explicit plot of factorial is shown in figure 8.1. I
extend a little bit that plot in figure 8.2, and I add some other related
functions. The complex map of factorial is shown in figure 8.3.

𝑦𝑦 = Factorial(𝑥)

5
𝑦 =𝑥

4
3

−1 (𝑥)𝑦2

orial= Fact

1
𝜈0

𝜈1

−3

−2

−1

𝜈0 𝜇 01/𝜇1 1𝑦 = Fac2

3

torial(𝑥 −1
)

𝑥

−1
−2
−3
𝜇1

−4
−5

https://mizugadro.mydns.jp/t/index.php/File:FactoReal.jpg
Fig. 8.2:

Factorial and related functions for real argument

91

𝑦
6
4
2

𝑣 =0
𝑢=0
𝑣 =0
𝑢=0
𝑣 =0

0

𝑢=
𝑣=0
𝑢 =0
𝑣=0
𝑢= 0
0

−2

0
=
𝑣 = 00
𝑢 = 0
𝑣 =
𝑢 =0
𝑣

−4
−6
−8

−6

−4

−2

0

2

4

6

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Factorialz.jpg
Fig. 8.3:

Complex map: 𝑢+i𝑣 = Factorial(𝑥+i𝑦)

For construction of superfunction of any transfer function, there is important question about its fixed points. The fixed points of factorial
are solutions 𝐿 of equation Factorial(𝐿) = 𝐿 . For real values of argument, the explicit plot of factorial is shown in figure 8.2 with thickcurve, 𝑦 = Factorial(𝑥). The fixed points correspond to intersections of
this curve with line 𝑦 = 𝑥, also shown in the figure. For comparison, thethin curves show functions Factorial−1 and 𝑧 ↦→ Factorial(𝑧)−1 . These
curves are added in order to remind, that Factorial𝑛 (𝑧), Factorial(𝑧 𝑛 )and Factorial(𝑧)𝑛 have pretty different meanings. Complex maps offactorial and ArcFactorial are shown in figures 8.3 и 8.4.
I had found no C++ implementations of factorial and arcfactorial forcomplex arguments in the literature. Mathematica software allows theevaluation, but does it a little bit slowly; the Maple software happened
to be not better [51], see also the appendix, chapter 22, section 2.
92

2

𝑣=

0.
6

0.8
𝑣=

𝑣 =1

𝑣 =1

4

1.6

.2

𝑣=

1.4

6

𝑣 =1

𝑣=

𝑦

.4
0
=

.8
𝑣 =2

𝑣
.2
0
=
𝑣

0 cut
8

𝑣 =0

𝑢=

3

𝑢=

−4

−2

𝑣 =−
1.2

−6

0

8
−0.

−1
.4

.2

3
=

𝑣=

−8

𝑣=

.6
1
−

𝑣 = −1

−6

1.8
−
𝑣=

𝑣=

−4

2.

−2 𝑣 = −2

𝑢=

2

4

6

𝑢

4
3.

𝑥

https://mizugadro.mydns.jp/t/index.php/File:AfacmapT800.png
Fig. 8.4: 𝑢+i𝑣 = ArcFactorial(𝑥+i𝑦)

For evaluation of superfunction and the abelfunction, the transfer function and/or its inverse should be evaluated many times, and the efficient(quick and precise) implementation is important. For this reason, theoriginal “complex double” procedures for factorial and arcfactorial aresuggested and loaded as
https://mizugadro.mydns.jp/t/index.php/Fac.cin andhttps://mizugadro.mydns.jp/t/index.php/Afacc.cin

These implementations are used to plot figures of this Chapter.
In figure 8.2, I show also line 𝑦 = 𝑥 and graphics 𝑦 = Factorial−1 (𝑥)and 𝑦 = Factorial(𝑥)−1 . Some extremal points of factorial are shown;the local minimum at point 𝑥 = 𝜈0 and local maximum at point 𝑥 = 𝜈1 ;Values of factorial in these points are denoted as 𝜇0 и 𝜇1 . Function
𝑦 = Factorial−1 (𝑥) has the branch point 𝑥 = 𝜇0 , and 𝜈0 is its value atthis point.

93

3

Regular iteration for factorial

The asymptotic expansion of superffactorial can be written as follows:

𝐹 (𝑧) = 𝐿 + exp(𝑘𝑥) + 𝑎2 exp(2𝑘𝑧) + 𝑎3 exp(3𝑘𝑧) + ..
= 𝐿 + 𝜀 + 𝑎2 𝜀2 + 𝑎3 𝜀3 + ..

(8.5)

where 𝜀 = exp(𝑘𝑧), for some constant increment 𝑘 and constant coefficients 𝑎 According to the general formulas (6.9), (6.10), (6.11), substitution of this expansion into the transfer equation
(8.6)

(︀
)︀
Factorial 𝐹 (𝑧) = 𝐹 (𝑧 +1)

gives the value of the increment. For Factorial’s fixed point 𝐿 = 2, I get(︀)︀
𝑘 = ln 3+2 Factorial′ (0) = ln(3−2 𝛾) ≈ 0.61278745233070836(8.7)

where 𝛾 = −Γ′ (1) ≈ 0.5772156649 is Euer constant. I set 𝑎0 = 2 and𝑎1 = 1; then the partial sum in expansion (8.5) is easy to program. I get𝑎2𝑎3

𝜋 2 + 6𝛾 2 − 18𝛾 + 6≈ 0.798731835172434541585621(8.8)=
12(3 − 5𝛾 + 2𝛾 2 )(︁= − 36 − 39𝜋 2 − 738𝛾 2 + 324𝛾 + 99𝜋 2 𝛾 − 60𝜋 2 𝛾 2 − 𝜋 4 + 24𝛾 5
)︁
3
2
3 2
4
+594𝛾 − 120𝜁(3)𝛾 + 48𝜁(3)𝛾 + 12𝛾 𝜋 + 72𝜁(3) − 204𝛾 /
(︁
)︁
2
3
4
5
144(−18 + 69𝛾 − 104𝛾 + 77𝛾 − 28𝛾 + 4𝛾 )(8.9)

≈ 0.5778809754764832358038

∞
∑︁

1
.
𝑧
𝑛
𝑛=1
We need 𝜁(3) ≈ 1.202056903 ; values of function 𝜁 at other arguments
are not used for evaluation of coefficients 𝑎. Approximate values ofcoefficients in expansion (8.5) are shown in table 8.1.
In equation (8.9), the Riemann zeta-function appears, 𝜁(𝑧) =For factorial, the increment 𝑘 > 0; expansion (8.5) gives good approximation 𝐹˜ (𝑧) at −ℜ(𝑧) ≫ 1. For other 𝑧 , I use the integer iterates; thesuperfunction

(︁

)︁
˜
𝐹 (𝑧) = lim Factorial 𝐹 (𝑧 −𝑛)𝑛𝑛→∞(8.10)For ℜ(𝑧) ∼ 1, it is sufficient to iterate expression (8.10) only few times,
in order to get 𝜀 of order of 0.1 or less. Then, some 15 terms in the
expansion (8.5) provide of order of 14 correct significant figures of 𝐹 (𝑧).
Let 𝑧3 be is real solution of equation 𝐹 (𝑧3 ) = 3; the evaluation gives𝑧3 ≈ −0.91938596545217788 ; then, I define

SuFac(𝑧) = 𝐹 (𝑧3 +𝑧)(8.11)

Function SuFac is also superfunction of factorial, and SuFac(0) = 3 .
94

Table 8.1: Coefficients 𝑎 and 𝑈 in the expansions (8.5) and (8.14)

I have displaced the argument of superfactorial, inorder to let it have integer value at zero. This issmallest integer, that is still
greater than the fixed point
𝐿 = 2.

𝑈𝑛−0.7987318351724345
0.6980641135593670−0.6339640557572815
0.5884152357911399−0.5538887519936520
0.5265479025985924−0.5041914604280215
0.4854529800293392−0.4694346809094714

9

7
6

𝑥)

8

𝑦 = Factor
ial(𝑥)𝑦https://mizugadro.mydns.jp/t/index.php/File:Sufact.png

𝑎𝑛0.7987318351724345
0.5778809754764832
0.3939788096629718
0.2575339580323327
0.1629019581037053
0.1002824191713524
0.0603184725913977
0.0355544582258062
0.0205859954874424

𝑦 = SuFac(

𝑛
2
3
4
5
6
7
8
9
10

Graphic of function SuFacof the real argument is
5
shown in figure (8.5), 𝑦 =4
SuFac(𝑥). For comparison,in the same figure, factorial
3
is shown, 𝑦 = Factorial(𝑥).
Graphic of factorial passes
2
through points with coor1
dinates (0,1), (1,1), (2,2),(3,6). Graphic of the superfactorial passes through−3 −2 −1 0 1 2 3 𝑥
points with coordinates (0,3), Fig. 8.5: 𝑦 = 𝑥! and 𝑦 = SuFac(𝑥) by (8.10),(8.11)(1,6), (2,720); the last point
is far away from the range of the figure.
Complex map of SuFac is shown in figure 8.6. This function has period

𝑃 =

(︀
)︀
2𝜋i
= 2𝜋i ln 3+2 Factorial′ (0) ≈ 10.253449681156 i (8.12)
𝑘This period is pure imaginary. A little bit less than two periods fit theheight of the upper map in figure 8.6.

95https://mizugadro.mydns.jp/t/images/d/d2/Superfactocomple1.png

𝑦
6

𝑢=2

𝑣 = −0.2

𝑣 =0

𝑢 = 1.84

𝑢=1
𝑢 = 1.2

𝑢=1

𝑣 = 0.2𝑢=2

𝑢 = 0.8856031944

2
0

𝑣 =0

𝑣 =0
𝑢 = 2.2

𝑣 =0

𝑣 =0

−2 𝑢 = 2

𝑢=1𝑣 = −0.2

−4
𝑣 =0

𝑣 =0

−6

𝑢 = 1.8𝑢=2−8−6𝑣 = 0.2

𝑦
2

−4

−2

𝑢 = 1.2

0

2

4

𝑢=1
6
𝑥

1

3

𝑣 =0

𝑢=0

𝑢=

4

𝑢=

𝑣 =0

𝑣=

2

2

0.8

𝑢=

𝑢=

𝑢=

𝑣 =1

1

𝑢 = 0.8856031944

1

2

3

4

5

https://mizugadro.mydns.jp/t/index.php/File:Sfaczoo300.png
Fig. 8.6:

𝑢+i𝑣 = SuFac(𝑥+i𝑦) and the zoom-in of the central part96𝑥

In the half-strips⃒⃒
𝑥 > 2, ⃒𝑦 + 𝑛|𝑃 |⃒ < 1, for integer 𝑛(8.13)

SuFac(𝑥 + i𝑦) has countable set of singularities. At the map, every
negative integer value of ℑ(SuFac(𝑥+i𝑦)), at the translation for unity
along the real axis, produces the singularity, due to the transfer equation(8.6). In order to show the structure of these singularities, the zoom-inof the map is shown at the bottom of figure 8.6. Outside the half-strips(8.13), factorial is regular; it approaches to the fixed points 𝐿 of factorial:
𝐿 = 2 in the left hand side and 𝐿 = 1 in the right hand side of the map.
The fast growth of factorial implies the slow growth of the inverse function, id est, growth of the abelfactorial. I denote this function withsymbol AuFac. Complex map of AuFac = SuFac−1 is shown in figure
8.7; this function is considered in the next section.

4

Abelfactorial

The asymptotic series for super factorial can be inverted. This gives theasymptotic expansion of abelfactorial (arcsuperfactorial) in vicinity of
the fixed point 𝐿 = 2:

(︃𝑁 −1)︃∑︁
1˜𝑈𝑛 (𝑧 −2)𝑛 + 𝒪(𝑧 −2)𝑁
𝐺(𝑧)
= ln
𝑘
𝑛=1

(8.14)

Parameter 𝑘 has the same meaning, as in expansion (8.5); its value isdetermined by (8.7). The first two coefficients 𝑈 are(8.15)

𝑈1 = 1
2

𝑈2 = −

2

𝜋 + 6𝛾 − 18𝛾 + 6≈ 0.7987318
12(3 − 5𝛾 + 2𝛾 2 )(8.16)The Mathematica routine InverseSeries allows to calculate these coefficients analytically, although the expressions for the highest 𝑈 are alittle bit long. The approximate values of these coefficients are shownin the right column of table 8.1.
For argument close to 2, the truncated series in (8.14) provides theprecise evaluation. For other values, the repeated application of therecurrent formula(︁)︁𝐺(𝑧) = 𝐺 ArcFactorial(𝑧) + 1
97(8.17)𝑦6

𝑢 = 1.6

𝑢 = 1.4

𝑢 = 1.8𝑣 = 0.4
𝑢 = 1.2

4
𝑣 = 0.62

𝑣 = 0.2

𝑣 = 0.8

𝑢=1
𝑣 =0

cut

0

−2 𝑣 = −0.6

𝑣 = −0.8
𝑢 = 1.2−4−6−8

𝑢 = 1.8−6

𝑢 = 1.6𝑣 = −0.6−4−2

𝑢 = 1.4 𝑣 = −0.4
024

𝑢 = 1.46𝑥

https://mizugadro.mydns.jp/t/index.php/File:AbelFacMapT.jpg
Fig. 8.7:

𝑢+i𝑣 = AbelFactorial(𝑥+i𝑦).

is used, until |𝑧 −2| becomes small. This algorithm determines the cut
line of the abelfctorial AuFac; the cut goes from 2 to −∞.
The abelfunction 𝐺 of factorial can be defined with−𝑛˜𝐺(𝑧) = lim 𝐺(Factorial(𝑧) + 𝑛𝑛→∞

(8.18)

then, AuFac = AbelFactorial = SuFac−1 , appears as

AuFac(𝑧) = 𝐺(𝑧) − 𝑧3 = 𝐺(𝑧) − 𝐺(3)(8.19)

where 𝑧3 = 𝐺(3) ≈ −0.91938596545218. This expression automaticallyprovides condition AuFac(3) = 0. In such a way,

(︁
)︁
𝑛
˜
AuFac(𝑧) ≈ AuFac𝑛 (𝑧) = 𝐺 ArcFactorial (𝑧) + 𝑛 − 𝑧3

98

(8.20)

al(𝑥)𝑦toricaFunction AuFac is plotted
F
c
r
A
𝑦=
in figure 8.8. For compari- 2
son, graphic of ArcFactorial
( 𝑥)
c
a
F
u
A
1
𝑦=
is also shown. Graphic of
ArcFactorial goes through 0
1
2
3
4
5
6
7 𝑥
points (1,1), (2,2), (6,3),(24,4), although the last −1
one is already out of range−2of the figure (and out ofrange of the page. Graphicof AuFac passes through https://mizugadro.mydns.jp/t/index.php/File:Aufact.png
Fig. 8.8: ArcFactorial and AuFac
points (3,0), (6,1), (720,2),
and the last one is not only
our of range of the page, but also out of room, were this Book is written,
and the next point, where AuFac takes the integer value, id est, (720!,
3) is far away from the visible part of our Universe.
Function AuFac grows very slowly. If the argument of this function
represents some physical quantity (distance, mass, charge, number ofatoms, etc.), measured in any reasonable units, then, there is no way tomake this quantity so big, that AuFac of it reaches 3.
While superfactorial and abelfactorial, id est, SuFac and AuFac, arealready constructed and implemented, one can use them to evaluate thenon-integer iterates of factorial. These iterates are matter of the next
section.

5

Iterates of factorial

With super factorial and abelfactorial, the 𝑛th iterate of factorial canbe written as follows:(︁)︁ℎ(𝑧) = Factorial (𝑧) = SuFac 𝑛 + AuFac(𝑧)𝑛(8.21)In this representation 𝑛 has no need to be integer. Figure 8.9 showsgraphic 𝑦 = Factorial𝑛 (𝑥) versus 𝑥 for various real 𝑛. Figure 8.9 should
be compared to figures 4.2, 4.5, 4.6, 4.7, 4.8, 4.13, 5.11, that show iterates
of other functions, that can be expressed through the special functions,known at least since century 20. There is no similar representation for
the abelfactorial. I hope, in century 21, some of new superfunctions,

99

9
8

𝑛 = 0.5𝑛=0
𝑛 = .4𝑛 = 0 .3
0.2
𝑛=𝑛 = 0 .1
0

𝑛→∞
𝑛=2
𝑛=1

𝑦

.1
0
−
𝑛 = 0.2−=
𝑛 −0.3𝑛 = −0.4𝑛=.5𝑛 = −07654

𝑛 = −1𝑛 = −2𝑛 → −∞

3
2
1
1

2

3

4

5

6

7

8

9

𝑥

https://mizugadro.mydns.jp/t/index.php/File:FacitT.jpg
Fig. 8.9:

𝑦 = Factorial𝑛 (𝑥) versus 𝑥 for various 𝑛

including SuFac, AuFac will be included in the handbooks on the special functions and implemented as built-in routines in the programming
languages; then, the difference between iterates of factorial and iterates
of other functions, mentioned in the previous chapters, will be even lesssignificant.
At 𝑛 = 1/2, formula (8.21) gives the half iterate of factorial. Map offunction ℎ = Factorial1/2 is shown in figure√ 8.10. This function can be! , used as logo of the Phys.
interpreted as “square root of factorial”
Dep. of the Moscow State University, shown in figure 2.4.
Concept of square root of factorial caused confusions and discussions.
Some colleagues did not want to see difference
between expiressions√
Factorial1/2 (𝑧) and Factorial(𝑧)1/2 , interpreting Factorial(𝑧) as equiv100

𝑦
6
4
𝑢=

−4

0

𝑢=

𝑣 =0

−2

𝑣 =8

𝑢=0

cut

𝑢=0

0

𝑣 =0

2

𝑢 = 0.4
𝑢 = 0.68𝑣 = −8𝑣 = 0.4

𝑢 = −8

−6
−8

−6

−4

−2

0

2

4

6

𝑥

https://mizugadro.mydns.jp/t/index.php/File:QFacMapT.jpg
Fig. 8.10:

𝑢+i𝑣 = Factorial0.5 (𝑥+i𝑦)

alent of Factorial(𝑧). In order to eliminate, en fin, this confusion,function ℎ = Factorial1/2 is considered here with more details.

√︀

The key for the verification of interpretation of function ℎ =analysis of the range of validity of relationℎ(ℎ(𝑧)) = Factorial(𝑧) = 𝑧!

√

! , is

(8.22)

Map of function ℎ2 in the left hand side of equation (8.22) is shown infigure 8.11. This figure should be compared to figure 8.3, that represents
the complex map of factorial.
In the right hand side of figure 8.11, complex map of the second iterateof function ℎ, id est, ℎ ∘ ℎ = ℎ2 , coincides with the map of factorial,
shown in figure 8.3. At least in the halfstrip ℜ(𝑧) > 1, |ℑ(𝑧)| ≤ 4, relation(8.22) holds. The resulting half iteration of factorial corresponds to the
101

𝑦
6
t

cu

4
2

𝑣 =0

0
−2
−4
−6
−8

−6

−4

−2

0

2

4

https://mizugadro.mydns.jp/t/index.php/File:QQFacMT.jpg

6𝑥2Fig. 8.11: 𝑢+i𝑣 = ℎ (𝑥+i𝑦).

intuitive expectations about this function.
Range of validity of relation (8.22) is limited with cut lines, shown infigure 8.11. Such cutlines are typical for non-integer iterates, if thesuperfunction cannot be represented through elementary function.
Factorial is not exotic, not an exception. With superfunctions, one canbuild-up non-integer iterates for other holomorphic functions too. More
examples are considered in the following chapters below.
Many transfer functions can be treated in a way, similar to that of this√chapter. One of them, namely, the exponential to base 𝑏 = 2, isconsidered in the next chapter.

102

Chapter 9
Exponent to base sqrt(2)𝑦=𝑦 = exp√
(
2 𝑥)
𝑥

𝑦
7
6
5
4
3
2
1
−5 −4 −3 −2 −1

0

1

2

3

4

5

𝑥

https://mizugadro.mydns.jp/t/index.php/File:ExpQ2plotT.png√
Fig. 9.1: Exponent to base 𝑏 = 2√

Exponent to base 2 happened to be first function treated with theregular iteration [61]. With this example (and for this function), themethods, described in chapter 6, were developed. In this chapter, thisexponent to this specific base is considered as transfer function 𝑇 .
Exponent to base√

2 can be expressed through the natural exponent:)︂(︂(︁ √ )︁
ln(2)𝑇 (𝑧) = exp√2 (𝑧) = exp ln( 2)𝑧 = exp
𝑧
(9.1)
2
Explicit plot of this function is shown in figure 9.1.
√
For exponential to base 𝑏 = 2, the fixed points 𝐿 = 2 and 𝐿 = 4 arenatural numbers. In figure 9.1, they correspond to intersections of curve
𝑦 = 𝑇 (𝑥) with the straight line 𝑦 = 𝑥.
103

𝑦𝑣 = 0.4
𝑢 = 0.4

𝑢=1

𝑢=2
𝑢=3

𝑢=0

𝑣 = −0.4

𝑣 = −1𝑣 = −2

−6

−4

−2

0

2

𝑣 =1

𝑣 =2
𝑣 =3

𝑣 = 0.2

𝑣 =0

𝑢 = 0.24

𝑢=0𝑣 = −0.2

6

2
0
−2
−4
−6
−8
−8

4

6

https://mizugadro.mydns.jp/t/index.php/File:ExpQ2mapT.png
Fig. 9.2:

𝑢+i𝑣 = exp√2 (𝑥+ i𝑦)Complex map of exponent to base 𝑏 =function is periodic; its period

𝑃 =

𝑥

√

2 is shown in figure 9.2. This

2𝜋i
2𝜋i
= (︁√ )︁ ≈ 18.1294405673 iln(𝑏) ln2

(9.2)

is pure imaginary. A little bit less than one period fits the height of map
in figure 9.2.

√

For the exponential to base 𝑏 = 2, the inverse function is logarithm tobase 𝑏; this logarithm can be interpreted as minus first iterate:

𝑇 −1 (𝑧) = log√2 (𝑧) = log𝑏 (𝑧) =
√

2
ln(𝑧)
ln(2)

(9.3)

Complex map of logarithm to base 𝑏 = 2 is shown in figure 9.3. The
dashed line marks the cut of the range of holomorphism; that cut runs
104

𝑣=

6

3

𝑣 =4

𝑦

4

𝑣=

2
0

2

𝑣=

𝑣 =9
𝑣 = −9

1

𝑣 =0

−2

𝑣=

−4

𝑢=3
𝑢=4

−6

𝑢=5

𝑣=

−1

𝑣=

−
2

−3

−8
−8

−6

−4

−2

0

𝑢=6
2

4

6

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Logq2mapT1000.jpg
Fig. 9.3:

𝑢+i𝑣 = log√2 (𝑥+ i𝑦)from zero along the negative part of the real axis. The jump at the cut
line is determined by the period (9.2) of the exponent:log√2 (𝑥 + i𝑜) − log√2 (𝑥 − i𝑜) = 𝑃 ≈ 18.1294405673 i(9.4)

for 𝑥 < 0. At the map, the levels 𝑣 = ℑ(log√2 (𝑥 + i𝑦)) = −9 and 𝑣 =ℑ(log√2 (𝑥+i𝑦)) = 9 are seen close to the negative part of the real axis.
Fixed points 𝐿 = 2 and 𝐿 = 4 of the exponent are also fixed points of
the logarithm; these points are seen in figures 9.1, 9.2 and 9.3.

√

This chapter describes the regular iteration of exponential to base 2
at the fixed point 𝐿 = 4. The next section describes the construction of
the superfunction, id est, the growing superexponential to this base.

105

Chapter 6 describes the constructionof iterates of a transfer function, thatare regular in vicinity of its fixed
point. Here, that method is used
for 𝑇 = exp√2 ; the superfunction
is constructed, that exponentially approaches the fixed point 𝐿 = 4 at large
negative values of the real part of theargument, and grows to infinity at thelarge positive argument. This function is denoted with SuExp√2,5 . The
last superscript at the name of thefunction indicates its value at zero,
SuExp√2,5 (0) = 5. In figure 9.4, curve
𝑦 = SuExp√2,5 is compared to that of𝑦 = exp√2 (𝑥). Below I construct anddescribe function SuExp√2,5 .

𝑦20

I use formula (6.2) of the asymptotic
expansion of superfuction 𝑓 :7𝑓 (𝑧) =

𝑁−1∑︁https://mizugadro.mydns.jp/t/index.php/File:Sqrt23uplot.jpg

Superfunction at fixed point 𝐿 = 4
19
18
17
16
15
14
13
12
11

8

Exp √

9

2,5 (𝑥)10𝑦 = Su

1

6
5

(︀
)︀
𝑎𝑛 e𝑛𝑘𝑧 +𝑂 e𝑁 𝑘𝑧

(9.5)

𝑛=0

)
(𝑥

4
3

√ 2

p
ex

Here I assume, that 𝑎0 = 𝐿 = 4 and2𝑦=𝑎1 = 1. Coefficient 𝑎1 could be cho1
sen arbitrary, but then other coefficients depend on 𝑎1 . Alteration of 𝑎1−3 −2 −1 0 1 2 3 𝑥
is equivalent of scaling of parameter Fig. 9.4: 𝑦 = SuExp√ (𝑥) by
2,5
𝜀 and displacement of the argument√of the superfunction. Then, formula (9.13) and 𝑦 = exp 2 (𝑥)(6.9) gives

)︁
(︁
√ )︁
𝑘 = ln exp𝑏 (4) = ln 4 ln( 2) = ln(2 ln(2)) ≈ 0.32663425997828 (9.6)(︁′This value determines periodicity of the superfunction; its period

𝑃 = 𝑃4 =2𝜋i
2𝜋i=≈ 19.236149042042854712 i𝑘ln(2 ln(2))(9.7)

I use equations (6.10),(6.11) or directly the transfer equation (which
gives the same results)√ to get coefficients 𝑎. While the transfer function
is exponent to base 2, the transfer equation can be written as follows:(︁√ )︁𝑓 (𝑧)𝑓 (𝑧 +1) =
2
106

(9.8)

Table 9.1:

Coefficients 𝑎 and 𝑈 in expansions (9.5), (9.15)

𝑛𝑎𝑛𝑈𝑛
1 1.0000000000000000
1.0000000000000000
2 0.4485874311952612 −0.4485874311952612
3 0.1903722467978068

0.2120891200549197
4 0.0778295765369683 −0.1021843675069717
5 0.0309358603057080
0.0496986830373718
6 0.0120221257690659 −0.0243075903261196
7 0.0045849888965617
8 0.0017207423310577
9 0.0006368109038799
10 0.0002327696003030
11 0.0000841455118381

0.0119330883965109−0.0058736976420089
0.0028968672871058−0.0014309081060793
0.0007076637148566

12 0.0000301156464937 −0.0003503296158730
13 0.0000106807458130

0.0001735756004664
14 0.0000037565713616 −0.0000860610119291
15 0.0000013111367785
0.0000426959089013
16 0.0000004543791625 −0.0000211930290682
17 0.0000001564298463

0.0000105244225996
18 0.0000000535232764 −0.0000052285174354
19 0.0000000182077863
0.0000025984499916
20 0.0000000061604765 −0.0000012917821121

Substitution of expansion (9.5) into the transfer equation (9.8) determines coefficients 𝑎𝑚 for 𝑚 > 1. In particular,

𝑎2
𝑎3 =

ln(2)/4
1 − 2 ln(2)(︁)︁
1−ln(2) ln(2)2 /12=

1−2 ln(2)−4 ln(2)2 +8 ln(2)3≈ 0.448587431195261(9.9)≈ 0.1903722467978067 (9.10)

Similar (but longer) expressions can be written for the other coefficients𝑎. The first column in table 9.1 suggests the approximate values ofcoefficients 𝑎 in expansion (9.5).
I use the truncation of the expansion (9.5), taking into account 𝑁 = 20
107

𝑦

𝑣 =0
𝑣=

0.4

0.
2

8

𝑢=4
𝑢=

2

𝑣 =0

0
−2

𝑢=

4.
2

4

𝑣=

−4

4 .4

𝑣=

6

2
0.
−

𝑢=4

−10

−8

−6

−4

𝑢=3

𝑣 =0

𝑢=

−8

𝑢=3

3.8

.6
𝑢 = 3.
4
𝑢 = 3.2

−6

−2

0

2

4

6

𝑥

8

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2figf45bT.png
Fig. 9.5:

𝑢+i𝑣 = 𝑓˜(𝑥−1.11520724513161 + i 𝑦) ;terms. This gives the primary approximation 𝑓˜ for superfunction 𝑓 :𝑓˜(𝑧) =

19∑︁

𝑎𝑛 exp(𝑛𝑘𝑧)(9.11)

𝑛=0

Approximation 𝑓˜ is shown in figure 9.5. In order to simplify the comparison with other maps, the argument of function in this figure isdisplaced for the real constant 𝑥45 ≈ −1.11520724513161; this constant provides approximate condition 𝑓 (𝑥45 ) ≈ 5 and the exact equality
SuEx√2,5 (0) = 5 for the superfunction constructed below and shown infigure 9.6.
The primary approximation (9.11) allows to plot the complex map ofsuperfunction in the left hand side of the complex plane. The residual at
the substitution of the approximation (9.11) into the transfer equation(9.8) becomes of order of rounding errors at ℜ(𝑧) < −2. The readers areinvited to plot this residual by themselves (preferably), or to look at itin the right hand side of figure 4 of the original publication [61].
108

𝑦

𝑢=3

𝑣 =0
𝑣=

0.4

0.
2

8

𝑣 =0

0
−2

𝑣=

−4

𝑣 =2

𝑢=

4.
2
𝑢=

2

𝑢=2

4 .4

𝑢=4

4

𝑣 =1

𝑣=

6

2
0.
−

𝑢=4

𝑢=2

𝑣 = −1

−10

−8

−6

−4

𝑢=3

𝑣 =0

.6
𝑢 = 3.
4
𝑢 = 3.2

𝑢=

−8

𝑢=3

3.8

−6

−2

0

2

4

6

𝑥

8

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2figf45eT.png
Fig. 9.6:

𝑢+i𝑣 = SuExp√2,5 (𝑥+i𝑦) by (9.13)

Superfunction 𝑓 appears as limit

𝑓 (𝑧) = lim exp𝑏

𝑛

𝑛→∞

(︁

)︁
𝑓˜(𝑧 −𝑛)(9.12)

This limit does not depend on the number 𝑁 of terms in the primary
approximation (9.11). Instead of 19 terms, one could choose anotherconstant. However, the more terms are taken into account, the fasterdoes the limit converge. Choosing 𝑁 = 19, I keep in mind the implementation complex double. For the approximation of 𝑓 with 14 significant
figures, it is sufficient to choose 𝑛 > ℜ(𝑧)+2.
It is convenient, when at zero the superfunction takes integer value. I
choose this value to be 5. This is smallest integer, that is still greaterthan the chosen fixed point 𝐿 = 4. I denote this superfunction withsymbol SuExp√2,5 and define it as follows:

SuExp√2,5 (𝑧) = 𝐹 (𝑧) = 𝑓 (𝑥45 + 𝑧)
109

(9.13)

where 𝑥45 ≈ −1.11520724513161 is real solution of equation 𝑓 (𝑥45 ) = 0.
Namely this constant is used for the displacement of the argument offunction 𝑓˜ in figure 9.5 in order of simplify the comparison with map offunction SuExp√2,5 shown in figure 9.6.
In the left hand side of the complex plane, functions 𝑧 ↦→ 𝑓˜(𝑧 + 𝑥45 )and SuExp√2,5 asymptotically coincide. This gives way to the precise
evaluation. In the whole complex plane, the superfunction can be approximated with any arbitrary precision; so, it should be qualified asexact solution. In such a way, the primary approximation 𝑓˜ provides
the exact solution for the superfunction SuExp√2,5 in the whole complex
plane.

√For the transfer function as exponential to base 𝑏 = 2, the regular
iteration at the fixed point 𝐿 = 4 gives the function 𝐹 = SuExp√2,5 thatis holomorphic in the whole complex plane. Graphic of this function isshown in figure 9.4. Complex map of function SuExp√2,5 is shown infigure 9.6.
For function 𝐹 = SuExp4,5 , value at zero is chosen a smallest integer,
which is still greater than 𝐿 = 5; so, 𝐹 (0) = 5. Then, function 𝐹 =SuExp4,5 can be interpreted as iterate of exponent with initial value 5:(9.14)

𝐹 (𝑧) = SuExp√2,5 (𝑧) = exp𝑧√2,u (5)where subscript u indicates, that the regular iteration is built-up at thehighest (“upper”) fixed point of the exponent.
In order to evaluate the non-integer iterates, the abelfunction is also
required; I mean, 𝐺 = 𝐹 −1 = AuExp√2,5 . This abelfunction is describedin the next section.

2

Abelfunction at fixed point 𝐿 = 4√

Superfunction 𝐹 = SuExp√2,4 of the exponent to base 2 is descried in
the previous section. This section considers the inverse function, 𝐺 =√ . Then, the combination of superfunction
AuExp√2,4 = 𝐹 −1 = SuExp−1
2,4
𝐹 and abelfunction 𝐺 allows to evaluate the non-integer iterates of thetransfer function 𝑇 = exp√2 by the general formula (2.14).
Expansion for the abelfunction 𝑔 = 𝑓 −1 can be obtained, inverting theasymptotics (9.5). This gives for the abelfunction 𝑔 expansion 𝑔˜ in thefollowing form:

exp(𝑘˜𝑔 (𝑧)) =𝑁−1∑︁𝑈𝑚 (𝑧 −𝐿)𝑛 + 𝑂(𝑧 −4)𝑁

𝑛=1

110

(9.15)

where 𝑈0 = 0; this implies, that at the fixed point 𝐿, function 𝑔 becomes
infinite. Then, according to expansion (9.5), we get 𝑈1 = 1 and 𝑈2 = −𝑎2 .
While coefficients 𝑎 are represented with exact expressions (with infinite
precision), Matgematica or Maple can calculate of order of 10 coefficients
𝑈 in expansion (9.15). Routine, that perform this inversion, is called
InverseSeries. In particular,

𝑈2 =
𝑈3 =

ln(2)/4
1 − 2 ln(2)(︁)︁
1+4 ln(2) ln(2)2 /24≈ −0.4485874311952612289 (9.16)

1−2 ln(2)−4 ln(2)2 +8 ln(2)3≈ 0.21208912005491969757 (9.17)

Approximate values of coefficients 𝑈 are shown in the right hand sidecolumn of table 9.1. The same coefficients can be obtained also independently on the expansion of superfunction, by substitution to expansion(︃𝑁 −1)︃∑︁
1𝑔˜(𝑧) = ln
𝑈𝑚 (𝑧 −𝐿)𝑛 + 𝑂(𝑧 −4)𝑁𝑘𝑛=1

(9.18)

into the Abel equation

𝑔(exp𝑏 (𝑧)) = 𝑔(𝑧) + 1(9.19)In order to fit the with of the screen (at the authomatic computation) or
the width of the page (at the calculation with paper and pen), getting𝑔˜(𝑧) by formyla (9.18) it worth to use the new variable 𝜁 = 𝑧−𝐿 = 𝑧−4.
Then, first coefficients 𝑈 in expansion (9.15) can be found even without
computer.
Truncated expansion 𝑔˜(𝑧) approximates 𝑔(𝑧) at |𝑧 − 4| < 2. For othervalues, the iterates with the Abel equation (9.19) can be used. Abelfunction 𝐺 = 𝐹 −1 , shown in figure 9.7, appears as limit of these iterates:

AuExp√(︁)︁
˜ log𝑏 (𝑧 −𝑛) + 𝑛 + 𝑥45
2,5 (𝑧) = 𝐺(𝑧) = lim 𝑔𝑛→∞(9.20)

where 𝑥45 ≈ −1.11520724513161 is the same constant, that appears in
the definition (9.13) of the superexponent 𝐹 = SuExp√2,5 , providing
𝐹 (0) = 5.
The inverse function of any non-trivial entire function has cut(s). The
abelexponent 𝐺 = AuExp√2,5 is not exception. The cut of the rangeof its holomorphism is shown in figure 9.7 with dashed line. This cut
goes along the real axis from minus infinity to the branch point 4. The
abelexponent has logarithmic singularity at this point. This singularity
corresponds to the exponential approach of the superesponent to the
fixed point, as the real part of the argument goes to minus infinity.
111

𝑢 = 3 .4

1

1

𝑣=

6

𝑢=4

8

𝑣=

𝑦

𝑢=

3

𝑣=

2

4
2

𝑣 =0

cut

0
−2

𝑢 = 4.4

−6

4.

−8
−10

−8

−6

−4

2

𝑢=

𝑢=

𝑢=
4

−4

−2

0

2

𝑣=

4

3

1 𝑢 = 3 .4
−
6

𝑥

8

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2figL45eT.png
Fig. 9.7:

𝑢+i𝑣 = AuExp√2,5 (𝑥+i𝑦) by formula (9.20)

The readers are invited to plot the map of the region, where the relation
below holds:

SuExp√(︁)︁
√2,5 AuExp 2,5 (𝑧) = 𝑧(9.21)and the map of range of validity of relation

AuExp√(︁
2,5

SuExp√)︁2,5 (𝑧) = 𝑧(9.22)√

Especially for the exponent to base 2, the algorithms of evaluation of
the growing superfunction and the abelfunction are loaded as
https://mizugadro.mydns.jp/t/index.php/F45E.cin andhttps://mizugadro.mydns.jp/t/index.php/F45L.cin

They are implemented in C++ as complex double functions of complex
double arguments. With these algorithms, the Reader can reproduce
figures of this chapter, even without downloading generators of thesefigures.
112

3Notations

The most of notations, used in this Book, are already introduced. Thissection suggests some kind of short overview of these notations. forthe transfer function (practically, it is any holomorphic function, forwhich the superfunction and abelfunctions are considered): 𝐹 for thesuperfunction, and 𝐺 = 𝐹 −1 for the corresponding abelfunction. These
notations are convenient, while it is clear, which transfer function is kept
in mind, and which of its superfunctions is denoted with 𝐹 and whichof its inverse 𝐺 is assumed.
However, these notations may cause confusions, if cited from other chapters. Functions 𝑇 , 𝐹 and 𝐺 may have (and actually have) different
meanings in different chapters. For this reason, in this chapter, I introduce also long names SuExp√2,5 = 𝐹 and AuExp√2,5 = 𝐺. The names
above have simple mnemonics. The first letter indicates, that it refers
to Superfunction or to Abelfunction. This letter is capitalised, following
the tradition of language Mathematica. The second letter indicates, thatthe biggest, “upper” fixed point of the transfer function is used. This
have sense, if the transfer function 𝑇 is real-holomorphic, and has real
fixed point(s), and one of them can be√qualified as maximal, upper. Thistakes place for the exponent to base 2. The following three charactersrefer to the name of the transfer function, id est, exp; again, its first
letter√ is capitalised. The subscript indicates the base of this exponent,𝑏 = 2 and value of this superexponent at zero; 𝐹 (0) = 5 and 𝐺(5) = 0.

4

Iterates of exponent to base 𝑏 =√2With superfunction 𝐹 = SuExp√2,5 and abefunction 𝐺 = AuExp√2,5 , the𝑛th iterate of transfer function 𝑇 = exp√2 can be expressed as follows:

𝑇𝑛(𝑧) = exp√𝑛2,u (𝑧) = SuExp√2,5(︁

𝑛+AuExp√)︁
2,5 (𝑧)(9.23)

This formula is valid at least for ℜ(𝑧) > 4. If one choose the upper (orlower) border at the cut of abelfuntion 𝐺 = AuExp√2,5 between 2 and 4,
the resulting iterate 𝑇 𝑛 (𝑧) is holomorphic in the whole compilex plane
except the cut along line 𝑧 ≤ 2. In figure 9.8, this 𝑇 𝑛 (𝑥) is shown versus
𝑥 for various real values of 𝑛. These iterates are defined at least for𝑥 > 2. For integer 𝑛, the curves can be extended also to 𝑥 ≤ 2.
For 𝑛 = 1/2, the complex map of function 𝑇 𝑛 (𝑧) by √(9.23) is shown infigure 9.9. This is half iterate of exponential to base 2:
113

12

𝑛=1
𝑛=0
.8
𝑛=
0 .6
𝑛=
0.4
𝑛=
0 .2

𝑛=

𝑦 =𝑏𝑥

13

𝑛=3
𝑛=2

𝑛→∞

𝑦

11

0

𝑛

0.4
−
𝑛=
0.6
−
𝑛 = 0.8
𝑛=− 1
) 𝑛=−

10

𝑦=

9

.2
0
=−

𝑥

8

5

𝑥
g
𝑏(
o
l
𝑦=
2
𝑛 = −2
)
𝑥
(
g
o
l
𝑦= 𝑏

4

𝑛 → −∞

76

𝑛 = −3

√

3

𝑏= 2

2
1
1

2

3

4

5

6

7

8

9

10

11

12

13

𝑥

https://mizugadro.mydns.jp/t/index.php/File:IterEq2plotT.jpg
Fig. 9.8:

𝑦 = exp√𝑛2,u (𝑥) by (9.23) for various 𝑛.

(︂
)︂
(︂
)︂
1
1
1/2
𝑇 1/2 (𝑧) = exp√2,u = 𝐹 +𝐺(𝑧) = SuExp√2,5 +AuExp√2,5 (𝑧) (9.24)
2
2
I remind, that symbol “u” in the subscript indicates, that for the regular iterate, the highest (“upper”) fixed point is used as asymptotics ofsuperfunction at infinity.
Abelfunction 𝐺, built up at the fixed point 𝐿 = 4, has logarithmic singularity at this point, and the corresponding cut; the jump at this cut
is constant; and this constant is just period of the superfunction 𝐹 . For
this reason, the iterates, even non-initeger, and in particular, the half
iterate by equation (9.24), are regular at the point 𝐿 = 4. However,the non-integer iterates have branch point at another fixed point of thetransfer function, namely, 𝐿 = 2. In figure 9.9, the corresponding cut ismarked with dashed line.
114

𝑦
8

𝑢 = −2
1
−
𝑢=

6

2

cut

0

𝑢=4
𝑢=6

𝑢=

𝑢=
1
𝑢=2

0

4

−2

𝑣 =6
𝑣 =4
𝑣 =2
𝑣 =0
𝑣 = −2𝑣 = −4

−4
−6
−8

−10

−8

−6

−4

−2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Esqrt2ite12mapT80.jpg
Fig. 9.9:5𝑢+i𝑣 = 𝑇 1/2 (𝑥+i𝑦) by formula (9.24).

Intermediate finish

Several results and tools for superfunctions, abelfunctions and non-integer
iterates are already presented above.
Here I make a view on the way already past and announce, what will
be in the future way, as it is shownin figure 9.10.
This chapter, above, deals with trans(︀√ )︀𝑧
fer function 𝑇 (𝑧) = 2 = exp√2 (𝑧);superfunction 𝐹 , abelfunction 𝐺 and
115

Fig. 9.10: View from the saddle pointiterates of the transfer function 𝑇 are described:

𝐹 = SuExp√2,5(9.25)𝐺 = AuExp√2,5(9.26)𝑇 𝑛 (𝑧) = 𝐹 (𝑛 + 𝐺(𝑧)) = exp√𝑛2,u (𝑧)(9.27)

Iterates by equation (9.27) are shown in figure 9.8. These iterates look
similar to iterates of other growing functions, in particular, those shownin figures 4.13 for iterates of the power function and 8.9 for iterates
of factorial. One may expect, at least for the real-holomorphic growing
functions, the similar iterates can be constructed in the similar way, even
if the superfunction and the abelfunction cannot be expressed throughthe special functions described in the textbooks of century 20. I mean,
with the regular iteration,
With the examples above, the Reader already understands, how can one
iterate a holomorphic function in such a way, that the number to iterates
has no need to be integer. These iterates greatly extends the tools,
available for scientific description of various processes, for approximation(fitting) of physical (and, perhaps, not only physical) dependences. The
non-integer iterates of exponential can be used, and, in particular, those√to base 2. The example of such a non-integer iterates is shown infigure 9.9. I believe, the Reader can easy plot other examples of theiterates of the exponential to this base, and also for some other valuesof base.
The results presented above may cause impression, that they finish theinvestigation of superfunctions, abelfunctions and non-integer interates,
and that, in future, one needs just to apply the regular iteration tovarious special functions. Actually, it is not a case.
For some functions, the regular iterates with asymptotic (6.2),(6.3) cannot be constructed, as the equations for the coefficients of the expansion
have no solution. One of these cases is considered in the next chapter
and qualified as “exotic”.

116Chapter 10
Exotic iteration and exp to base e1/e𝑦 = 𝑦 = √𝑦 = 𝜂 𝑥(𝑥 2) 𝑥https://mizugadro.mydns.jp/t/index.php/File:Expe1eplotT.jpg

𝑦
7
6
5
4
3e
2
1
−5 −4 −3 −2 −1Fig. 10.1:

𝑦 = 𝑏 for 𝑏 = 𝜂 = e
𝑥

1/e

0

1

2

e

3√

4

5

𝑥

≈ 1.44466786 and 𝑏 = 2 ≈ 1.41421356

The regular iteration by (6.2)-(6.12) looks as general method of construction of superfunctions. However, in some cases, it cannot be applied asis. In particular, the asymptotic expansion (6.2) becomes invalid, if thederivative of the transfer function at the fixed point is unity:

𝑇 (𝐿) = 𝐿,

𝑇 ′ (𝐿) = 1(10.1)In this case, expression 𝑇 ′ −1 = 𝑇 ′ (𝐿)−1 in denominator of fraction in
the right hand side of equation (6.10) becomes zero; and the coefficientsin expansion (6.2) loss their meaning. Exponential to base 𝑏 = e1/e ≈1.44466786 is an example of such an exotic transfer function. It is shownin figure 10.1 and considered in this chapter.
117

1

Exotic iterate

The coincidence, that the derivative in equation(10.1) becomes unity, can be qualified as exotic.
This determines the title of the chapter and that
if this section. Condition (10.1) is not only exotic
possible, (see figure 10.2), but, first, I consider case(10.1).

Fig. 10.2: Exotics

The general formulas of chapter 6 fail for the special case 𝑇 ′ (𝐿) = 1, and Henryk Trappmann expected, that for this casethe precise evaluation of superfunction is very difficult, if at all. In order
to convince him, that it is doable, we had to write the paper [80]. Partof that publication is described below.
This section considers the case, when the derivative of the transfer function 𝑇 in the fixed point 𝐿 is unity. Then, formulas of section 6 for the
regular iteration cannot be applied as is.
Calculation of iterates of a function is simpler, if its fixed point is zero.
If the fixed point 𝐿 of transfer function 𝑇 is not zero, then, the superfunction 𝐹 can be represented in the following form:

𝐹 (𝑧) = 𝑓 (𝑧) + 𝐿(10.2)

Then, for function 𝑓 , we get𝑓 (𝑧 +1) = 𝐹 (𝑧 +1) − 𝐿 = 𝑇 (𝐹 (𝑧))−𝐿 = 𝑇 (𝐿+𝑓 (𝑧)) − 𝐿(10.3)

We may define

𝑇new (𝑧) = 𝑇 (𝐿 + 𝑧) − 𝐿(10.4)and interpret this 𝑇new as new transfer function, and 𝑓 is superfunction for it. Below, the subscript new is omitted. This is equivalent toassumption

𝑇 (0) = 0

(10.5)

Let the Taylor expansion of the transfer function have the followingform:

𝑇 (𝑧) = 𝑧 + 𝑣𝑧 2 + 𝑤𝑧 3 + ..

(10.6)

where 𝑣 ̸= 0. For such a transfer function, it is difficult (and, perhaps, impossible) to built-up any superfunction, that exponentially approaches
118zero at infinity. But it is possible to construct the superfunction, thatdecays, roughly, as the inverse proportional function. Let𝑎 𝑏ℓ 𝛼ℓ2 +𝛽ℓ+𝛾𝑓 (𝑧) = + 2 +
+ ..
𝑧 𝑧
𝑧3

(10.7)

where 𝑎, 𝑏, 𝛼, 𝛽 and 𝛾 are constants, and ℓ = ln(𝑧). Below I assume,that coefficients 𝑣 and 𝑤 of the expansion (10.6) are known, and show,
how to calculate coefficients 𝑎 and 𝑏 of the asymptotic (10.7).
For the displacement of the argument 𝑧 ↦→ 𝑧 + 1, expressions in theasymptotic representation (10.7) are transformed in the following way:

(︂
)︂−1
1
1
1
1
1
1
1
↦→
=
(10.8)
1+= − 2 + 3 + ..
𝑧𝑧 +1 𝑧
𝑧
𝑧 𝑧
𝑧
(︂
)︂−2
1
1
1
2
3
1
1
→
↦
=
−
+
+ .. (10.9)
1+=𝑧2(𝑧 +1)2𝑧2
𝑧
𝑧2 𝑧3 𝑧4
(︂ (︂
)︂)︂
1
ℓ = ln(𝑧) ↦→ ln(𝑧 +1) = ln 𝑧 · 1 +
𝑧
(︂
)︂
1
1
1
= ln(𝑧) + ln 1 += ℓ + − 2 + .. (10.10)𝑧
𝑧 2𝑧
Using these preparations, the left hand side of the transfer equation𝑓 (𝑧 +1) = 𝑇 (𝑓 (𝑧))(10.11)can be written as follows:

(︂
)︂
𝑎
𝑎
𝑎
𝑏ℓ + 𝑏/𝑧2
𝛼ℓ2 +𝛽ℓ+𝛾𝑓 (𝑧 +1) = − 2 + 3 +
1
−
+
+ ..
𝑧 𝑧
𝑧
𝑧2
𝑧
𝑧3
)︁ 1 (︁)︁𝑎
1 (︁2= + 2 −𝑎+𝑏ℓ + 3 𝑎+𝑏 − 2𝑏ℓ + 𝛼ℓ +𝛽ℓ+𝛾 + .. (10.12)𝑧 𝑧
𝑧The right hand side of equation (10.11) becomes

(︂
)︂2
(︁ 𝑎
)︁3
𝑎 𝑏ℓ 𝛼ℓ2 +𝛽ℓ+𝛾𝑎 𝑏ℓ
𝑇 (𝑓 (𝑧)) = + 2 ++𝑣·+ + .. + 𝑤+ .. ..
𝑧 𝑧
𝑧3
𝑧 𝑧2
𝑧
)︁ 1 (︁
)︁
𝑎
1 (︁
2
2
3
= + 2 𝑏ℓ+𝑣𝑎 + 3 𝛼ℓ +𝛽ℓ+𝛾 +2𝑣𝑎𝑏ℓ+𝑤𝑎 +..
(10.13)𝑧 𝑧
𝑧
We should equalize the right hand sites of equations (10.12) and (10.13).
Coefficients at 𝑧1 coincide automatically. Coefficients at 𝑧12 gives−𝑎 = 𝑣𝑎2
119(10.14)

Equalisation of coefficients at 𝑧13 gives

𝑎 + 𝑏 − 2𝑏ℓ = 2𝑣𝑎𝑏ℓ + 𝑤𝑎3(10.15)

Variable ℓ = ln(𝑧) depends on 𝑧 , but equation (10.15) should be valid
for various 𝑧 . So, we get the two equations:−2𝑏 = 2𝑣𝑎𝑏

(10.16)

which, at 𝑏 ̸= 0, gives the same, as (10.14), and also

𝑎 + 𝑏 = 𝑤𝑎3(10.17)

Solving these two equations, I get

𝑎=

−1
𝑣

𝑏=

,

−𝑤 1
+
𝑣3
𝑣

(10.18)One can add more terms in the expansion (10.7), and, in the similarway, calculate coefficients 𝛼, 𝛽 , 𝛾 and even higher, assuming, that the
expansion (10.6) of the transfer function is known. In such away, thecoefficients of the asymptotic expansion (10.7) of the superfunction 𝑓
are determined by the coefficients of expansion of transfer function 𝑇 .
The asymptotic solution (10.7) allows evaluation of the transfer function
at large 𝑧 . Then, as in the case of the regular iteration, the asymptotic
solution can be extended to the whole complex plane (except the cut
lines) with the transfer equaltion, applying one of the two formulas below:

𝑓 (𝑧) = 𝑇

𝑛

(︁

)︁
𝑓 (𝑧 − 𝑛)(10.19)(︁(10.20)or𝑓 (𝑧) = 𝑇−𝑛)︁
𝑓 (𝑧 + 𝑛)

in order to make the argument of the superfunction large, bringing it to
the range, where the truncated asymptotic expansion (10.7) provides therequired precision. The choice of one of the formulas (10.19) or (10.20)
determines the position of the cut lines of the resulting superfunction.
The direction of the cut lines can be changed also with replacement ofℓ = ln(𝑧) to ℓ = ln(−𝑧). In order to get unique solution, we need tochoose (and postulate) the asymptotic behavior of the function we wantto construct. In the following section, the method above is applied to120

exponential to base e1/e , and then, in the following chapters, to otherfunctions.
In principle, even with two terms in equation (10.7), one can get thecamera-ready maps of the superfunction and related functions. Perhaps, this is sufficient for the application in physics and other sciences,where the precision of measurements is usually less than 14 decimal
digits. The precision can be infinitely improved with modification of
the argument of the primary approximation 𝑓 with equations (10.19) or(10.20), increasing the number 𝑛 of iterates.
However, for the numerical tests, it worth to calculate several coefficientsof expansion of superfunction. This reduces the time of evaluation andimproves the precision. For various transfer functions, that are holomorphic in vicinity of the fixed point and have expansion beginning with
(10.6), some tens of coefficients of the expansion (10.7) can be calculated.
Examples of the numerical tests for such expansions are presented below.

2

Exponent to base exp(1/e)

Let us apply the formalism of exotic iterations above to the exponential
to base 𝑏 = 𝜂 = exp(1/e) ≈ 1.44466786 . This section deals with function
𝒯 by

𝒯 (𝑧) = 𝜂 𝑧 = exp𝜂 (𝑧) = exp(ln(𝜂) 𝑧) = exp(𝑧/e)(10.21)

Explicit plot of this transfer function is shown in figure 10.1 with thick√curve. For comparison, the thin curve shows the exponent to base 2,
considered above in chapter 9; this curve is borrowed from figure 9.1.
The curves at this figure look close; at small and negative values of theargument, they almost overlap. However, the deviation is significant atlarge positive values of the argument, and this determines the pretty
different behaviour of the corresponding superfunctions.
Complex map of the transfer function exp𝜂 is shown in figure 10.3. This
map looks similar to the map at figure 9.2; the only period is slightly
different. For the base 𝜂 , this period

𝑃exp𝜂 = 2 𝜋 i e ≈ 17.0794684453 i(10.22)

It is amassing, that the simple formula (10.22) combines the 3 fundamental constants, : 𝜋 , i and e. In addition, the Henryk base 𝜂 = exp2 (−1) =
121

𝑦

𝑣 =2
𝑣 =3

𝑣 =1

𝑢=0

𝑣 = 0.44𝑣 = 0.2

6𝑣 =0

𝑢 = 0.2

𝑢 = 0.4

𝑢=1

𝑢=2
𝑢=3

𝑢=0

𝑣 = −0.4

𝑣 = −1𝑣 = −20𝑣 = −0.2

2

−6

−4

−2

0

2

−2
−4
−6

4

6

https://mizugadro.mydns.jp/t/index.php/File:Expe1emapT1000.jpg
Fig. 10.3:𝑥

𝑢+i𝑣 = exp𝜂 (𝑥+i𝑦)

exp(1/e) ≈ 1.44466786101 also is mathematical constant. A little bit
less than one period 𝑃exp𝜂 fits the range of map in figure 10.3.
For transfer function 𝒯 = exp𝜂 , the inverse function is 𝒯 −1 = exp−1
𝜂 =log𝜂 . Complex map of this function is shown√in figure 10.4. This map
looks similar to the map of logarithm to base 2, shown in chapter 9 atfigure 9.3. At that map, the mesh of isolines is a little bit more dense;
for example, all the line 𝑢 = 6 happen to be in the range of the map,
while at the map at figure 10.4 the only part of this line is seen; as forlevel 𝑣 = ±9 it happen to be beyond the cut line and, therefore, not seenin the map.
Value 𝑏 = 𝜂 is maximal real base, at which the exponent still has atleast one real fixed point. Below, for this exponent, the superfunction
is constructed.
122

𝑣=

𝑣=

𝑣 = −8
𝑣

𝑢=1
𝑢=2
𝑢=3

7
−
=

𝑢=5

−4

𝑣=

𝑢=4

−2

0

−

−1

2

−3

−6

𝑣=

𝑣=

−6

1

𝑣 =0

cut

𝑣=

−4

2

𝑣=

𝑣 =8

0
−2

7

−6

2

𝑣=

6

4

3

6

𝑣=

𝑣 =4

𝑦

2

4

6

https://mizugadro.mydns.jp/t/index.php/File:Loge1emapT1000.jpg
Fig. 10.4:

3𝑥

𝑢+i𝑣 = log𝜂 (𝑥+i𝑦)

Superexponent to base 𝜂 = e1/e = exp2(−1)

This case had been considered by request from Henryk Trappmann. He
exopectes, that for the exponential to base

𝜂 = exp(1/e) = exp2 (−1) ≈ 1.44466786101(10.23)

I cannot construct the efficient (fast and precise) algorithm of evaluation.
The similar opinion had been expressed in century 20 by Peter L. Walker[24].
In order to convince Henryk, we had to write the special article for thejournal Mathematics of Computation [80]. Some formulas and figuresfrom that publication are repeated below.
123Let the transfer function 𝒯 be defined with equation 10.21 of the previous section; let(︀)︀
𝒯 (𝑧) = 𝜂 𝑧 = exp𝜂 (𝑧) = exp ln(𝜂) 𝑧 = exp(𝑧/e)(10.24)In order to apply the formalism of exotic iteration from the previous
section, I define the new, “displaced” transfer function, as it is suggested
by equation (10.4); for the initial transfer function 𝒯 by (10.24). To
distinguish these transfer functions, I use slightly different fonts for theletter T in the names. The new (displaced) transfer function(︀)︀
𝑇 (𝑧) = 𝒯 (𝑧 + e)−e = exp (𝑧 +e)/e − e = exp(𝑧/e+1)−e(10.25)For this transfer function,

𝑇 (0) = 0

(10.26)𝑇 ′ (0) = 1(10.27)𝑇 ′′ (0) = 2 𝑣 = 1/e(10.28)𝑇 ′′′ (0) = 6 𝑤 = 1/e2

(10.29)(10.30)...
𝑇 (𝑛) (0) = 1/e𝑛−1(10.31)In the last formula, in the left hand side, the parenthesis in the superscript indicates not a number if iterate, but number of derivative;function 𝑇 is differentiated 𝑛 times.
For this case, the formalism of exotic iteration, described in the precious
section, can be applied as is. Expansion (10.6) can be written as follows:

𝑇 (𝑧) = 𝑧 +For 𝑣 =1 2
1
𝑧 + 𝑧 3 + ..
2e6e

(10.32)

1
1
and 𝑤 = 2 , formulas (10.18) give2e6e𝑎 = −2e ≈ −5.43656365691809e1
𝑏 = − + 3 ≈ 1.604598172578777
3 8e(10.33)(10.34)

Then, the primary approximation 𝑓 for superfunction (10.7) has thefollowing form:

1 (︁1 )︁ ln(±𝑧)˜
𝑓 (𝑧) = − + 1 +
+ ..
𝑧
e
𝑧2
124

(10.35)More terms of this expansion can be calculated analytically, especially,
if some Mathematica or Maple are used. In principle, the truncatedseries in (10.35), even with two terms, approximates the superfunction
and can be used for its definition. For |𝑧| > 100, such a representation
gives few correct decimal digits and can be used to plot the complex
maps.
For the accurate representation of superfunction, more terms in the expansion, similar to (10.35), should be calculated. The asymptotic expansion can be written as follows:−2e𝐹˜ (𝑧) =
𝑧

(︃
1+

(︀
)︀
𝑀
∑︁
𝑃𝑚 − ln(±𝑧)
𝑚=1

(3𝑧)𝑚

(︁

+ 𝒪 ln(±𝑧)/𝑧)︁𝑀 +1)︃(10.36)

where

𝑃𝑚 (𝑥) =𝑚∑︁

𝑐𝑚,𝑛 𝑥𝑛(10.37)𝑛=0

One may choose the first coefficient 𝑐0,0 = 1; then, 𝑃0 (𝑡) = 1. Other
coefficients 𝑐 are determined by the substitution of the expansion (10.36)
into the transfer equation

(︁
)︁
˜
˜
𝐹 (𝑧 +1) = exp 𝐹 (𝑧)/e + 1 − e
The first 5 polynomials 𝑃 are shown below:(10.38)𝑃1 (𝑡) = 𝑡
𝑃2 (𝑡) = 𝑡2 + 𝑡 + 1/2
5
7
5
𝑃3 (𝑡) = 𝑡3 + 𝑡2 + 𝑡 +
2
2
10
13
45
53
67
𝑃4 (𝑡) = 𝑡4 + 𝑡3 + 𝑡2 + 𝑡 +
3
6
10
60
77
101
83
653
2701
𝑃5 (𝑡) = 𝑡5 + 𝑡4 +𝑡3 + 𝑡2 +
𝑡+
12
6
4
60
1680
The superfunction 𝐹 appears as limit
𝐹 (𝑧) = lim 𝑇 𝑛 (𝑓 (𝑧 − 𝑛))𝑛→±∞(10.39)(10.40)(10.41)(10.42)(10.43)where function 𝑓 is just truncation of the asymptotic expansion in(10.36) at some positive integer 𝑀 , and 𝑛 is chosen positive or negative, dependently on the sign in the argument of logarithm in formula(10.36).
125

It is convenient to deal with functions, that have integer value at zero;so, I use the translation of the argument in order to define iteration tet𝜂and the growing sperexponential SuExp𝜂,3 with the following formulas:tet𝜂 (𝑧) = 𝐹1 (𝑧) = 𝑓 (𝑧 +𝑥1 )(10.44)

SuExp𝜂,3 (𝑧) = 𝐹3 (𝑧) = 𝑓 (𝑧 +𝑥3 )(10.45)where constants 𝑥1 ≈ 2.798248154231454 and 𝑥3 ≈ −20.28740458994004
are chosen to provide relations 𝐹1 (0) = 1 and 𝐹3 (0) = 3.
Function tet𝜂 corresponds to the upper sign go the ± in formuas (10.36),(10.43), while function SuExp𝜂,3 by (10.45) refers to the lowerst sign. In
such a way, 𝐹1 and 𝐹3 are pretty different functions, and it is difficult
to express one of them through another.
In formulas (10.44) and (10.45), for the two superfunctions 𝐹1 and 𝐹3 , alittle bit longer names tetexp(1/e) and SuExpexp(1/e) are suggested. Theselong names simplify the identification at the use from other chapters of
this Book (and also from other publications), where the base 𝑏 may havevarious values, and not necessary 𝑏 = 𝜂 .
I repeat the meanings of the names suggested. tet𝜂 , indicates, that thisrefers to tetration. For any tetration (to any base 𝑏), I assume condition
tet𝑏 (0) = 1.
The shoice of value at zero of the growing super exponential 𝐹3 is not
so obvious. In publications [61, 80] (and not only there), the value atzero is chosen as minimal integer, that is still larger, than the real fixed
point. Such a choice has sense, while the only one transfer function
without parameters is considered, or if value of its parameter is fixed,
and no continuity properties with respect to this parameter are analysed.
However, if we treat base 𝑏 as parameter, then, the choice of integer value
at zero leads to discontinuity with respect to base 𝑏. For the case, if sucha dependence will be requested, the notation SuExp𝜂,3 ; is used; the value
at zero is indicated as the additional subscript.
For the real argument, functions 𝐹3 and 𝐹1 are shown in figure 10.5.
Complex maps of these superfunctions are shown in figure 10.6. These
functions are pretty different. In order to release the notation 𝐹 with
subscripts for other functions, used in other chapters (and in other publications), I give them the special names SuExp𝜂,3 = 𝐹3 and tet𝜂 = 𝐹1 .
126

)︁
(︁
𝑦 = ℜ 𝐹1 (𝑥 +i𝑜)𝑦8𝑦 = SuExp𝜂,3 (𝑥) = 𝐹3 (𝑥)

6
4

𝑦 =e
𝑦 = tet𝜂 (𝑥) = 𝐹1 (𝑥)

2
0
0

2

4

6

8

10

12

14

16

18

𝑥

https://mizugadro.mydns.jp/t/index.php/File:E1eplot8.png
Fig. 10.5:

Two super-exponentials to base 𝑏 = exp(1/e)

Symbol tet refers to tetration; in the following chapters, it is definedalso for other values of base.
Along the real axis, function SuExp𝜂,3 infinitely grows at the positive
values of the argument, and approaches its limiting value e at large
negative values. The function approaches to the same value in the mostof the complex plane, except the strip along the positive direction od
the real axis.
Function tet𝜂 , at large values of the argument, approaches its limiting
value 𝜂 almost everywhere, but has logarithmic singularity at −2, going
to infinity at this point. Tetration to any base has this singularity;
it follows from the additional condition tet𝑏 (0) = 1 and the transfer
equation, that in vicinity of the real axis can be rewritten in the following
form:

(︁

)︁
tet𝑏 (𝑧) = log𝑏 tet𝑏 (𝑧 +1)(10.46)For iterates of exponent to base 𝜂 , we need not only the superfunctions, but also the corresponding abelfunctions. These abelfunstions are
considered in the next section.

127https://mizugadro.mydns.jp/t/index.php/File:E1esuma8.jpg

𝑦𝑣 = 0.2

𝑢 = 2.812

𝑢 = 2.6108𝑣 = 0.4
𝑢 = 2.4

𝑣 = 0.4

6

𝑢=3

4
2

𝑢 = 3.2

𝑣 =0

𝑣 =0

0

huge values

𝑢=2

−2
−4
−6

𝑣 = −0.4−8−10

𝑢 = 2.8 𝑢 = 2.6𝑣 = −0.2−12−14−8 −6 −4 −2

0

2

4

6

10 12 14 16 18 20 22 24 26 28 𝑥

8

https://mizugadro.mydns.jp/t/index.php/File:E1etetma8.jpg

𝑦𝑢 = 2.8

𝑢 = 2.612𝑣 = 0.2

𝑣 = 0.4

10
8 𝑢=3
6
4
2

𝑢=2

0

𝑣 =0

𝑢 = 2.2−2

𝑢 = 2.4

−4
−6
−8
−10

𝑣 = −0.4−12−14

𝑢 = 2.8−8 −6 −4 −2

𝑣 = −0.2

𝑢 = 2.6
0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 𝑥

𝑢+i𝑣 = SuExp𝜂,3 (𝑥+i𝑦) = 𝐹3 (𝑥+i𝑦) by (10.45), at the top, and𝑢+i𝑣 = tet𝜂 (𝑥+i𝑦) = 𝐹1 (𝑥+i𝑦) by (10.44), at the bottom.
Fig. 10.6:

128

4

Abelexponent to Hernyk base

Historically, the exponent to base 𝜂 = exp(2/e) is first function, forwhich the exotic iteration had been applied [80]. The two superfunctions
for this exponent are specified in the previous section. Here, I describe
the inverse functions, id est, the two abelfunctions of the exponent tobase 𝜂 .
Let 𝐺1 = 𝐹1−1 and 𝐺3 = 𝐹3−1 . Maps of these functions are shownin figure 10.7. For evaluation of functions 𝐺1 and 𝐺3 , the asymptotic
representations are used:15ln(±𝑡) 2 ∑︁ 𝑛𝑔(𝑧) = 𝑔± (𝑧) =+ +𝑐𝑛 𝑡 + 𝑂(𝑡16 )3
𝑡 𝑛=1(10.47)

where 𝑡 = (𝑧 − e)/e. The coefficients 𝑐 can be found inverting the expansion for the superfunction, and also by substituting expansion (10.47)
into the Abel equation

(︁

)︀)︁
𝑔(𝑧) + 1 = 𝑔 exp 𝑔(𝑧)/e(︀(10.48)The truncation of the series, id est, omitting of 𝑂 in the right hand sideof (10.47), provides the algorithm for evaluation of the abelfunction 𝑔with at least 15 significant figures, while |𝑡| < 1/2. For larger values,
the argument of 𝑔 should be transformed, using the equation (10.48).
The inverse functions 𝐺1 = 𝐹1−1 and 𝐺3 = 𝐹3−1 can be defined, adding
the corresponding constants,

ate𝜂 (𝑧) = 𝐺1 (𝑧) = 𝑔(𝑧) − 𝑔(1) ≈ 𝑔(𝑧) − 3.029297214418(10.49)where the upper sign in (10.47) is used, and

AuExp𝜂,3 = 𝐺3 (𝑧) = 𝑔(𝑧)−𝑔(3) ≈ 𝑔(𝑧)+20.0563555297533789 (10.50)
where the lower sign in (10.47) is used.
The additional names ate and AuExp are introduced in order to simplify
identification and referencing to these functions from other chapters.
Representations (10.49) and (10.50) provide relations 𝐺1 (1) = 0 and
𝐺3 (3) = 0, as it is supposed to be for the inverse functions of 𝐹1 and𝐹3 . The numerical tests confirm, that in wide ranges of values 𝑧 , therelations

𝐹1 (𝐺1 (𝑧)) = 𝑧 , 𝐹3 (𝐺3 (𝑧)) = 𝑧 , 𝐹1 (𝐺1 (𝑧)) = 𝑧 , 𝐺3 (𝐹3 (𝑧)) = 𝑧
129(10.51)https://mizugadro.mydns.jp/t/index.php/File:E1egi4.jpg

𝑦𝑣 = 0.212108

𝑢 = 196𝑣 =1

4𝑢 = 18.22

𝑢 = 18cut

0

𝑢 = 18.4𝑣 =0−2𝑣 = −1−4−6

𝑢 = 19−8−10𝑣 = −0.2−12−14−8 −6 −4 −2

0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 𝑥

https://mizugadro.mydns.jp/t/index.php/File:E1eti4.jpg

𝑦complicated structure

1210

𝑢 = −38 𝑣 =06

𝐺1 (𝑥+i𝑦) ≈ −3

𝑢 = −3𝑢 = −2

4
2

𝑣 =0

0

complicated structure−2𝑢 = −2−4−6−8

𝑢 = −3𝑣 =0𝐺1 (𝑥+i𝑦) ≈ −3

𝑢 = −3−10−12−14−8 −6 −4 −2

0

2

4

6

8

complicated structure
10 12 14 16 18 20 22 24 26 28 𝑥

𝑢 + i𝑣 = 𝐺3 (𝑥 + i𝑦) = AuExp𝜂,3 (𝑥 + i𝑦) by (10.50), top map,and 𝑢+i𝑣 = 𝐺1 (𝑥+i𝑦) = ate𝜂 (𝑥+i𝑦) by (10.49), map at the bottom

Fig. 10.7:

130hold with 14 significant figures. The readers are invited to plot the mapof the agreement function(︂|𝐹 (𝐺(𝑧)) − 𝑧|
𝐴(𝑧) = − lg
|𝐹 (𝐺(𝑧))| + |𝑧|)︂(10.52)adding appropriate subscripts and exchanging 𝐹 ↔ 𝐺, as numericalcheck of equalities in statement (10.51). Levels of 𝐴(𝑥+i𝑦) in the 𝑥, 𝑦plane indicate, where the algorithm works almost without loss of precision, id est, close to the maximal precision allowed for the complex
double variables.

5

Iterates

With function 𝐹3 and 𝐺3 , one can express the iterates of exponential tobase exp(1/e). For 𝑇 (𝑧) = exp(𝑧/e),

𝑇u𝑛 (𝑧) = exp𝑛𝜂,u (𝑧) = 𝐹3 (𝑛+𝐺3 (𝑧)) = SuExp𝜂,3(︁)︁
𝑛+AuExp𝜂,3 (𝑧) (10.53)

Here, the subscript 𝑢 indicates, that the this iterate is holomorphic at
the highest, upper range of the real values of the argument, and, inparticular, in the vicinity of the half-line 𝑧 > e.
In the representation (10.53), neither argument 𝑧 nor number of iterate
has need to be real. For 𝑛 = 1/2, the complex map of the half iterationof exponential to base exp(1/e) is shown in figure (10.9).
For real values of argument 𝑇u𝑛 (𝑥) versus 𝑥 for various values of number
𝑛 of iteration is shown in figure 10.8. The thick curves correspond to
the integer values of 𝑛. The non-integer iterates are holomorphic at
least in some vicinity of half-line 𝑥 > e. The integer iterates can beextended to −∞ for positive 𝑏 or to the closest vertical asymptotic
of the corresponding logarithm for negative number of iteration. The
bisector of the First quadrant of the coordinate plane corresponds to theidentity function.
The non integer iterates by (10.53) are shown in figure 10.8 with thinlines. These lines cannot be extended beyond the fixed point e; this isbranch point of the non-integer iterates. These lines remain in the range𝑥 > 𝜂 , 𝑦 > 𝜂 ; in particular, they cannot be holomorphically extended into
the third quadrant.
The non-integer iterates by formula (10.53) have cut line along the real
axis from −∞ to e ≈ 2.71; in order to shown this cut, figure 10.9
131

𝑛=1
𝑛=
𝑛 = 0 .6
𝑛 = 0 .40𝑛 = .2
0

9

𝑛=5
𝑛=3
𝑛=2

𝑛 → +∞

𝑦

7

.2
0
−
=
𝑛 0.4
−
=
𝑛

6

1
𝑛=−

5

𝑛 = −2𝑛 = −3𝑛 = −58

4
3
e 𝑛 → +∞
2𝑛 → −∞

10
0𝑛 → −∞

1

2

e3

4

5

6

7

8

9

𝑥

https://mizugadro.mydns.jp/t/index.php/File:E1eiterT.jpg
Fig. 10.8:𝑛(𝑥) by (10.53) versus 𝑥 for various 𝑛𝑦 = exp𝜂,urepresent the complex map of the half iterate, 𝑛 = 1/2. This cut istraced with dashed line and labeled with symbol cut.
As other complex maps of other iterates of a real-holomorphic function,
the map at figure 10.9 is symmetric with respect to reflection from thereal axis,*
1/2*
exp1/2𝜂,u (𝑧 ) = exp𝜂,u (𝑧)(10.54)In order to remind this symmetry, I plot the maps, placing the real axis
at the centre of the figure. However, sometimes, there is not enoughspace at the page, then I plot the only upper part of the map. I hope,
the Reader has enough imagination, that allows him or her to consider
the imaginary mirror, and see in this imaginary mirror the function withinverted sign of the its imaginary part. With hopes for such imagination,
in Figure 10.10, I show the iterates of the exponent to base 𝜂 for various132

0
𝑢 ==120𝑢𝑢=

𝑢 = 50
7
𝑢= 0
100

𝑦
14
12

𝑣=

10

00

𝑣 = 50

𝑢=
𝑢= 0
1
𝑢=3

𝑣 =0

1
−
𝑣=

4

𝑣 =1

15

6

9
𝑣= 6
𝑣=

8

−2

𝑣 = 20𝑣 = 10𝑣 =0𝑣 = −10𝑣 = −20−4𝑣 = −50

20

cut - - - - - - -

−6
−8
−10
−12
−14
−8 −6 −4 −2

0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 𝑥

https://mizugadro.mydns.jp/t/index.php/File:E1eghalfm3.jpg
1/2
Fig. 10.9: 𝑢+i𝑣 = exp𝜂,u (𝑥+i𝑦) by (10.53) for 𝑛 = 1/2

number of iterate 𝑛 in a similar way, as the case 𝑛 = 1/2 is shown in
Figure 10.9. However, the only halts of the maps are shown in Figure
10.10.

6

Not only exp

Exponential to base 𝜂 = exp(1/e) = exp2 (−1) is not only transfer
function, that have derivative unity at its fixed point, 𝑇 ′ (𝐿) = 1. Someof such transfer functions can be treated with the method of exotic
iteration, described in this chapter.
One more example is considered in the next chapter, referring to theelementary transfer function 𝑇 (𝑥) = zex(𝑥) = 𝑧 exp(𝑧). It is treated ina pretty similar way, as exp𝜂 .
Then, in the following chapters, even more exotic cases are treated, when𝑇 ′ (𝐿) = 1 and 𝑇 ′′ (𝐿) = 0 also can be treated. However, I try to go stepby step. So, open the next chapter and read about iterates of zex.

133

𝑢=

𝑢=

2

1

0.4

𝑢=

𝑢=

2

1
𝑢=

0
𝑢 = .2
0 .4

𝑢=

𝑢=2

𝑢=1

0
𝑢=

𝑢=2

𝑢=1

𝑢=0

−1
𝑢=

−2
𝑢=

3
𝑣=

𝑢= 3
𝑢
𝑢 == 1 2
0

3

𝑣 =2

𝑣 =3
𝑢=1

𝑢=2

𝑢=0

𝑢 = −10 cut−3 −2 −1

0

1

2

𝑣=

𝑣 =0

𝑛 = −0.1

𝑣 =2

𝑣 =1

𝑣 =3
𝑣 =2

𝑣 =1

𝑣 =1
𝑣3 = 0𝑥

𝑣 =3
𝑣 =2

𝑣 =0

cut

𝑢=2

3

1

𝑣 =1

𝑢=1
𝑢=0
𝑢 = −1
2
𝑢=−

𝑣 =0
𝑣 =4

3

𝑣 =2

𝑣 =1

0 cut
𝑦

𝑛 = 0.1

𝑣=

𝑢=2
𝑢=1
0
𝑢=
−1
𝑢= 2
−
𝑢=

𝑛 = 0.2

2
𝑣=
𝑣 =1

𝑣 =0
2

0 cut
𝑦

1

3
𝑢=
2
=2
𝑢
0
𝑣 =1
𝑢 == −1𝑢𝑣 = 0 cut𝑣=
4 𝑛 = −0.5 𝑢
𝑣=
=
3
𝑢= 1
𝑢=− 0
𝑣 =2
1
𝑢 = −2𝑣 =1
𝑣 = 0 cut𝑣 =4
𝑛 = −0.2
𝑣 =3

𝑣=

𝑢=

𝑛 = 0.5

1

2

𝑣 =0
3

0.2
𝑢=

0

2

𝑣 =1

3

𝑢=

2

2

𝑣=

2
=
𝑢 =1 0𝑢 𝑢=𝑛 = −0.8

0 cut
𝑦

3

cut

𝑛 = 0.8

1

3

2
𝑣 =1
𝑣 =0

3

𝑣=

2

𝑣=

1

𝑣 =0

𝑛 = −0.9
𝑢=

3

0 cut
𝑦
3

cut

2

𝑣=

𝑣=

𝑛 = 0.9

𝑣=

𝑣 =8

𝑣=

1

2
𝑣 =1

𝑣 =0

3𝑢=0
2

𝑣=

𝑢=

2
𝑢=

1
0
𝑦

0
𝑢 = .4
0 .6
𝑢=
1

𝑢=

0.2

𝑛 = −1

3

2

𝑛=1

𝑣=

3

𝑢=

𝑦

cut
−3 −2 −1

0

1

2

3

𝑥

𝑛
Fig. 10.10: 𝑢+i𝑣 = exp𝜂,u (𝑥+i𝑦), complex maps of iterates of exponent

to base 𝜂 by equation (10.53) for various 𝑛

134

Chapter 11
LambertW and zex
𝑦

e

2
1
−3

−2

𝑦 = zex(𝑥)

0

−1

−1/e

1

2 𝑥

https://mizugadro.mydns.jp/t/index.php/File:ZexPlot.png
Fig. 11.1:

𝑦 = zex(𝑥) = 𝑥 exp(𝑥)This chapter deals with function ArcLambertW,

ArcLambertW(𝑧) = ArcProductLog(𝑧) = zex(𝑧) = 𝑧 exp(𝑧)shown in figure 11.1. The long name ArcLambertW is equivalent to even longer name ArcProductLog. This name looks
so long, as “Logistic map” or “dihidrogen monoxide” shown infigure 11.2. Formally, the name is correct; the inverse function of ProductLog can be called ArcProductLog in analogy
with arcsin or ArcTetration. But name zex is shorter.
Complex map of function zex is shown in figure 11.3.
135

H
Fig.

(11.1)
OH

11.2:

Dihidrogenmonoxide

𝑦

𝑢=0

5
4

𝑣 =0

3

𝑢=0

1
0

𝑣 =0

𝑢 = −0.22

𝑣 = 16𝑣 = −16
𝑢 = 16𝑢 = −16

−1
−2

𝑢=0

−3
−4

𝑣 =0

−5
−6
−6

𝑢=0
−5
−4
−3
−2
−1
0
1
2
3
4
5
https://mizugadro.mydns.jp/t/index.php/File:ZexD6mapT100.png

𝑥Fig. 11.3: 𝑢+i𝑣 = zex(𝑥+i𝑦) by (11.1)

Short name zex is created from the first three characters in the righthand side of equation (11.1). Hope, this long explanation helps to avoid
confusions. In the chapter name, I keep the indication to LambertW,
because it is used in many publications and algorithmic languages. As
the negative iterates are allowed, if one can iterate zex, one can iteratealso LambertW. Function zex is simpler than LambertW, so, I consider
zex first.
Consideration in this chapter is very similar to that of the previous
one; the similar exotic iterates are constructed. I hope, the reader canperform the calculus for iterates of zex in the same way, as the deduction
is performed in the previous chapter for the exponent to base 𝜂 = e1/e .
On the other hand, I remember the old rule:136

If at your presentation, at the first desk you see Hideki Yukawa andSofia Kovalevskaya, and at the last desk you see some Bart Simpson
and Shoko Okudaira, then you should address to Bart and Shoko. In
this case, you may hope, that Hideki and Sofia understand at least themain idea of your talk.
Adopting that rule to the Book, here I repeat the deduction of theprevious section for function zex. However, I would like the Readers atleast to try to make this deduction by themselves, using the Book only
to verify the results; then the Readers will be able to do the same also
for other functions (including exotic ones).

1

Holomorphic zex

For real argument, function zex by (11.1) is shown in figure 11.1. At
negative values of the argument, zex is negative, and it is positive forpositive argument. At large negative values of the argument, zex decreases, reaches its local minimum −1/e at −1, then grows up. The
graphic passes through point (0,0); zero is fixed point. At zero, its
derivative is unity; so the iterates of zex are qualified as exotic; the noninteger iterates are not regular at the fixed point. Here, it may be agood moment to go a little bit back, to the regular iteration, in order toremember, what happens, for example, at the right hand side of equation (6.10), when the derivative 𝑇 ′ of the transfer function at the fixed
point becomes zero.
Complex map of function zex is shown in figure 11.3. This function isentire, it has no singularities. Map of the inverse function

zex−1 = ArcZex = zex−1 = LambertW(11.2)is shown in figure 11.4. This function has cut, along the negative partof the real axis. This cut is marched with dashed line.
Function LambertW can be defined as solution 𝐹 of the differential
equation𝐹 (𝑧)(11.3)(1 + 𝐹 (𝑧)) 𝑧with additional condition 𝐹 (0) = 0, where the contour of integration goesalong the imaginary axis from zero to the imaginary part of argument 𝑧 ,𝐹 ′ (𝑧) =

137

𝑣 =1

𝑦
6
4

𝑣=

2

1 .8

𝑣=

.2
0
=
𝑣

2

𝑣 =0

0 cut

2
𝑣 =−

𝑢=

−2

𝑢=

−6

𝑢=1
−6

−4

.4
−2

𝑣=

𝑣 = −1

1
𝑢=
1 .2

−4

𝑣 =−
0.2

0 .8

0

2

4

−0
.4

𝑥

6

https://mizugadro.mydns.jp/t/index.php/File:LambertWmap150.png
Fig. 11.4:

𝑢+i𝑣 = ArcZex(𝑥+i𝑦) = LambertW(𝑥+i𝑦)

and then, parallel to the real axis, to the value 𝑧 . Solving the equation(11.3), the Reader may verify, that the inverse function of the solution
is zex. Note, that the similar contour of integration is used in chapter 5
to define function Tania by equation (5.3). Indeed, function LambertW
is related to the Tania function: (5.3):(︁)︁
LambertW(𝑧) = Tania ln(𝑧) − 1(11.4)As the imaginary part of the logarithm is limited to the −𝜋, 𝜋 range, inthis expression, the argument of the Tania function is always inside thestrip along the real part, seen in the complex map of the Tania function
in figure (11.4). Function LambertW is in some sense simpler, thanfunction Tania: it has only one cut line (and one branch point −1/e),
138

while the Tania functions has two.
I could not find good implementation complex double for function LambertW in C++ (in which I have good plotters of complex maps). For
this reason, I use the expansions listed below.
For small values of argument, LambertW can be expanded as follows:

LambertW(𝑧) = 𝑧∞∑︁(𝑛+1)𝑛−1
𝑛=0= 𝑧 −𝑧 2 +

𝑛!

(−𝑧)𝑛

3𝑧 3 8𝑧 4 125𝑧 5 54𝑧 6 16807𝑧 7−+−++ .. (11.5)
2
3
24
5
720

The series converges at |𝑧| < 1/e ≈ 0.367879 . With 48 terms, at |𝑧| ≤0.2, the truncated sum provides at least 16 correct decimal digits.
The expansion at the branch point can be written as follows:𝑡2 )︁
𝑡2 11𝑡3 43𝑡4
769𝑡5
221𝑡6
+
= −1 + 𝑡 − +−+−LambertWe2e
372
540 17280
8505
8
9
10
7
1963𝑡
226287557𝑡
5776369𝑡
680863𝑡−+−+ ..
(11.6)+
43545600 204120 37623398400 1515591000(︁ −1

The expansion can be used for approximation of LambertW while |𝑡| <1, id est, while the argument of LambertW is close to − exp(−1) ≈−0.367879 .
For large values of |𝑧|, using notations 𝐿 = ln(𝑧) and 𝑀 = ln2 (𝑧) the
expansion of LambertW(𝑧) can be written as follows:𝑀
𝑀 (−2 + 𝑀 )+𝐿
2𝐿2
𝑀 (6 − 9𝑀 + 2𝑀 2 )+
6𝐿3
𝑀 (−12 + 36𝑀 − 22𝑀 2 + 3𝑀 3 )+
12𝐿4
𝑀 (60 − 300𝑀 + 350𝑀 2 − 125𝑀 3 + 12𝑀 4 )+
60𝐿5
𝑀 (−120 + 900𝑀 − 1700𝑀 2 + 1125𝑀 3 − 274𝑀 4 + 20𝑀 5 )+120𝐿6(︁ 𝑀 )︁7+𝑂(11.7)
𝐿
LambertW(𝑧) =

𝐿−𝑀 +

139

where the effective parameter of expansion happens to be 𝜀 = ln2 (𝑧)/ ln(𝑧);at |𝜀| ≪ 1, the asymptotics (11.7) can be used for the evaluation ofLambertW. Here, as usually, the upper superscript after the function
indicates the number of iterations, and the upper superscript after theargument and the closing parenthesis indicates the power that is assumed to be evaluated after the evaluation of the function. However,ln−1 (𝑧) = exp(𝑧) should not be confused with 1/ ln(𝑧), nor ln2 (𝑧) =ln(ln(𝑧)) should be confused with ln(𝑧)2 , and so on.
The above asymptotics leads to the efficient (fast and precise) algorithm
for evaluation of LambertW. The C++ implementation is loaded as

https://mizugadro.mydns.jp/t/index.php/LambertW.cin.
With functions zex and LambertW = zex−1 , the superfunction of zex andits abelfunction can be implemented. The superfunction is consideredin the next section.

2

SuZex

Superfunction for the transfer function zex is solution 𝐹 of the transfer
equation(︁)︁𝐹 (𝑧 +1) = zex 𝐹 (𝑧)(11.8)For some integer 𝑀 > 1, search for the asymptotic expansion of thesolution in the following form:(︂𝐹 (𝑧) = 𝐹𝑀 (𝑧) + 𝑂

ln(±𝑧)𝑀 +1
𝑧 𝑀 +2)︂(11.9)

where
𝑀 𝑃
∑︁

1 12 ln(±𝑧) 1
𝐹𝑀 (𝑧) = − +
+
𝑧
𝑧2
𝑧 𝑚=2
𝑃𝑚 (𝑧) =𝑚∑︁

𝑐𝑚,𝑛 (−𝑧)𝑛

(︁
𝑚

)︁
ln(±𝑧)
𝑧𝑚(11.10)(11.11)𝑛=0

The coefficients 𝑐 can be found by substitution of expansion (11.9) into
the transfer equation (11.8). This can be done with the Mathematica
code below:
140

zex[z_] = z Exp[z];Foo[z_] = -1/z + a Log[z]/z^2
Soo = Series[Foo[z+1]-zex[Foo[z]], {z,Infinity,3}]
Eoo = Coefficient[Soo,1/z^3]
Ao = Extract[Solve[Eoo==0, a], 1]F2o[z_] = ReplaceAll[Foo[z], Ao]
F20[z_] = F2o[z] + (a Log[z]^2 + b Log[z] + c)/z^3
S2o = Series[F20[z+1] - zex[F20[z]], {z,Infinity,4}]
S20 = ReplaceAll[S2o, Log[1/z] -> -L]
E2o = Coefficient[S20, 1/z^4]
E22 = Coefficient[E2o, L^2]
A1 = Extract[Extract[Solve[E22==0, a], 1], 1]
E2A = ReplaceAll[E2o, A1]
E21 = Coefficient[E2A, L]
B1 = Extract[Extract[Solve[E21==0, b], 1], 1]
E2B = ReplaceAll[E2A, B1]
C1 = Extract[Extract[Solve[E2B==0, c], 1], 1]F3o[z_] = ReplaceAll[F20[z], {A1, B1, C1}]
F30[z_] = F3o[z]+(a Log[z]^3+b Log[z]^2+c Log[z]+d)/z^4
S3o = Series[F30[z+1] - zex[F30[z]], {z, Infinity, 5}]
S30 = ReplaceAll[S3o, Log[1/z] -> -L]
E3o = Coefficient[S30, 1/z^5]
E33 = Coefficient[E3o, L^3]
A3 = Extract[Extract[Solve[E33==0, a], 1], 1]
E3a = ReplaceAll[E3o, A3]
E32 = Coefficient[E3a, L^2]
B3 = Extract[Extract[Solve[E32==0, b], 1], 1]
E3b = ReplaceAll[E3a, B3]
E31 = Coefficient[E3b, L]
C3 = Extract[Extract[Solve[E31==0, c], 1], 1]
E3c = ReplaceAll[E3b, C3]
D3 = Extract[Extract[Solve[E3c == 0, d], 1], 1]F4o[z_] = ReplaceAll[F30[z], {A3, B3, C3, D3}]
F40[z_] = F4o[z] +(a Log[z]^4+b Log[z]^3+c Log[z]^2+d Log[z]+e)/z^5
S4o = Series[F40[z+1] - zex[F40[z]], {z, Infinity, 6}]141Such a calculus leads to the following asymptotics:
1
1
1 3
−5 2
3
−7
2
1 12 ℓ −1
4 ℓ + 4ℓ − 6
8 ℓ + 16 ℓ + 8 ℓ + 48
𝐹 (𝑧) = − + 2 +
+
𝑧 𝑧
𝑧3
𝑧4
−1 4
13 3
−17 2
23
−707
16 ℓ + 48 ℓ + 32 ℓ + 48 ℓ + 4320
+
𝑧5
−77 4
37 3
−83 2
1121
−1637
1 5
32 ℓ + 384 ℓ + 64 ℓ + 96 ℓ + 1728 ℓ + 8640
+
𝑧6
−1 6
87 5
−205 4443 3−1619 215427
−274133
64 ℓ + 640 ℓ + 384 ℓ + 384 ℓ + 1152 ℓ + 17280 ℓ + 1209600+
𝑧6(︂ 7 )︂

ℓ𝑧8(11.12)where ℓ = ln(−𝑧).

𝑦For some number 𝑀 of terms taken into
account, the superfunction 𝐹 can be expressed through its 𝑀 th asymptotic asfollows:(︁)︁
𝐹 (𝑧) = lim zex 𝐹𝑀 (𝑧 −𝑛)
𝑛

𝑛→∞

9
8
7

(11.13)

6

The resulting 𝐹 does not depend on thenumber 𝑀 of terms taken into accountin the primary approximation. However, the convergence of the limit forlarger 𝑀 is faster.
In order to simplify the comparison ofdifferent representations of the superfunction, it is convenient to define themisplaced function

SuZex(𝑧) = 𝐹 (𝑧1 +𝑧)

5
4
e

3
2

𝑦 = SuZex(𝑥) 1
𝑦 = 𝑥e𝑥−2 −1
0
1 𝑥(11.14)−1/e

https://mizugadro.mydns.jp/t/index.php/File:SuZexPlot511T.jpg+𝑂where 𝑧1 ≈ −1.1259817765745026 is solution of equation 𝐹 (𝑧1 ) = 1. This def- Fig. 11.5: 𝑦 = SuZex(𝑥)inition gives a way to evaluate the superfunction of zex. The complex double implementation in C++ isloaded as https://mizugadro.mydns.jp/t/index.php/SuZex.cin

For real values of argument, the explicit plot of function SuZex is shownin figure 11.5 with thick line. For comparison, the thin line shows the
142

𝑦
5
4
3
2

𝑣 =0

𝑢 = 0.40

𝑢 = 0.2

1

−1
−2
−3

𝑢

𝑢=0

−4
−5
−6
−6

𝑣=

0.2
−
=

.2
0
−

−5
−4
−3
−2
−1
0
1
2
3
4
5
https://mizugadro.mydns.jp/t/index.php/File:SuZexD1mapT.png
Fig. 11.6:𝑥

𝑢+i𝑣 = SuZex(𝑥+i𝑦)

zex function, id est, 𝑦 = 𝑧 exp(𝑧). These curves cross at point (1, e) andin vicinity of point (1.4, 6.2).
For real values of argument, SuZexp is positive monotonously growing
function. At −∞, it approaches zero, as the asymptotic representation(11.10) prescribes.
Then, the curve passes through point (0, 1), and then - through point(1, e). At this point it grows a little bit slower, than function zex, but
soon overdoes the zex, showing very fast growth; this growth is faster,than growth of any exponential.
The same behaviour can be seen also at figure 11.6, that represents thecomplex map of function SuZex. zex is entire function, id est, holomorphic in the whole complex plane. The inverse function, shown in figure
11.6, has cut. This inverse function is considered in the next section.
143

𝑦
8

𝑣 = 0.66𝑣=

𝑢 = 1.4

2

−2

𝑣 =0
1.6

cut

0

2
0.

𝑣=

𝑢=

4

𝑢=

2
0.

2

−

−4
−6

𝑣 = −0.6

−8
−10

−8

−6

−4

−2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:AuZexMapT.jpg
Fig. 11.7:

3

𝑢+i𝑣 = AuZex(𝑥+i𝑦)

AuZex

Complex map of function AuZex = SuZex−1 , id est, abelfunction for
the transferfunction zex, is shown in the right hand side of figure 11.7.
This section describes properties of function AuZex.
The asymptotic expansion for the abelfunction for the transfer function
AuZex can be obtained inverting the expansion of function SuZex, described in the previous section. However, one may consider as well the
Abel equation for the abelfunction 𝐺 :𝐺(zex(𝑧)) = 𝐺(𝑧) + 1(11.15)The asymptotics of solution 𝐺 = AuZex can be expressed with
𝑁
∑︁
−1 1
𝐺(𝑧) ≈+ ln(𝑧) +𝑏𝑛 𝑧 𝑛 + ..
𝑧
2
𝑛=0
144

(11.16)

Coefficients 𝑏𝑛 for 𝑛 > 1 can be found substituting this expansion into
the Abel equation (11.15). This asymptotics provides the precise approximation for small values of 𝑧 ; at large values, the inverse of theAbel equation should be applied. For some fixed 𝑀 , let
𝑀
∑︁
−1 1
𝐺𝑀 (𝑧) ≈+ ln(𝑧) +𝑏𝑛 𝑧 𝑛𝑧2
𝑚=0(11.17)and let(︁)︁
AuZex(𝑧) = lim 𝐹𝑀 zex (𝑧) + 𝑛
𝑛→∞
(︁
)︁
𝑛
= lim 𝐹𝑀 LambertW (𝑧) + 𝑛−𝑛𝑛→∞(11.18)

Coefficient 𝐶0 is chosen in such a way, that AuZex(1) = 0; then, therelation(︁)︁
SuZex AuZex(𝑧) = 𝑧(11.19)holds in wide range of values of 𝑧 , except the negative part of the realaxis. With Mathematica software, the coefficients 𝑏 of the asymptotic
expansion (11.17) can be calculated by the code below.
zex[z_] = z Exp[z];S[k_, L_] = Sum[a[k, m] L^m, {m, 0, k}]
F[K_, z_, L_] = Sum[S[k, L]/z^(k + 1), {k, 0, K}]
Series[zex[F[4,z,L]] - F[4, z+1,L+Log[1+1/z]], {z,Infinity,3}]a[0,0] = -1;
Series[zex[F[4, z, L]] - F[4, z+1, L+Log[1+1/z]], {z,Infinity,3}]a[1, 1] = 1/2; a[1, 0] = 0;
Simplify[Series[zex[F[5,z,L]] - F[5,z+1, L+Log[1+1/z]], {z,Infinity,4}]]n = 2;
s[n]=Series[zex[F[n+3,z,L]]-F[n+3,z+1,L+Log[1+1/z]],{z,Infinity,n+2}];
For[k = 0, k<=n,k++,m=n-k;a[n,m] = ReplaceAll[a[n, m],So1[Coefficient[s[n] L,L*L^m] == 0, a[n,m]]];Print[n, Space, k, Space, m, Space, a[n, m] ] ]n = 3;
s[n]=Series[zex[F[n+3,z,L]]-F[n+3,z+1,L+Log[1+1/z]],{z,Infinity,n+2}];
For[k = 0, k<=n,k++,m=n-k;a[n,m] = ReplaceAll[a[n, m],So1[Coefficient[s[n] L,L*L^m] == 0, a[n,m]]];Print[n, Space, k, Space, m, Space, a[n, m] ] ]and so on for higher 𝑛. The first 9 coefficients are copypasted below:
145

𝑛

𝑏𝑛

approximation of 𝑏𝑛

1
2
3
4

−1/6
1/16
−19/540
1/48≈ −0.1666666666666666667≈ 0.0625≈ −0.0351851851851851852≈ 0.0208333333333333333

5−41/4200≈ −0.0097619047619047619
637/103680≈ 0.00035686728395061728
7 18349/3175200 ≈ 0.005778848576467624−443/80640≈ −0.005493551587301587
8(11.20)9 55721/21555072 ≈ −0.002585052835824441
The readers are invited to plot maps of ranges of validity of relations

SuZex(AuZex(𝑧)) = 𝑧
AuZex(SuZex(𝑧)) = 𝑧(11.21)(11.22)

These maps can be considered as confirmation, verification of the deduction above.

4

Iterates of zex

With functions SuZex and AuZex, described in the previous sections,the iterates of function zex can be expressed as follows:(︁)︁zex (𝑧) = SuZex 𝑛 + AuZex(𝑧)𝑛(11.23)As usually, the number 𝑛 of iterate has no need to be integer; it can bereal or even complex.
Iterates 𝑦 = zex𝑛 (𝑥) versus 𝑥 for different 𝑛 are shown in figure 11.8.
The integer iterates are shown with thick lines.
The readers are invited to check the relation
𝑛

zex

(︁

)︁
zex (𝑧) = zex𝑚+𝑛 (𝑧)𝑚(11.24)and describe the range of validity in simple terms. At least, this relation
should hold in some vicinity of the positive part of the real axis for valuesof the parameters.
Iterates of function zex are shown in figure 11.8. They look similar
to iterates of other fast-growing functions with real fixed point. The
146

0 .1

𝑛=

𝑛=

𝑛=

0 .2

.3
𝑛=0

𝑛=1

11

𝑛=2

𝑦

0

𝑦 = zex(𝑥)

10
9
8
7

𝑦=

𝑥

𝑛

.1
0
=−

0.2
−
=

6

𝑛

5

.3
0
−
𝑛=

4
3
2

𝑦 = LambertW(𝑥)1𝑛 = −1𝑛 = −2
1

2

3

4

5

6

7

8

9

10

11

𝑥

https://mizugadro.mydns.jp/t/index.php/File:ZexIteT.jpg
Fig. 11.8:

𝑦 = zex𝑛 (𝑥) by (11.23) versus 𝑥 for various 𝑛iteration keeps the unity derivative in this point, so, all the curves infigure 11.8 approach the fixed point (id est, to the origin of coordinates)with unity derivative. Id est, with angle 45 degrees to the abscissa axis.
This property takes place also for other transfer functions with unity
derivative at the fixed point. One of such functions is considered in thenext chapter.
I hope, the Readers can plot by themselves the complex maps of noninteger iterates of function zex. The codes supplied to the figures above
have the complex double implementation of SuZex and AuZex.

147

Chapter 12Sin, super sin and iterates of sin
Not all exotic iterations can be constructed with formulas of the previous
sections. In the previous two sections, the transfer functions 𝑇 with fixed
points 𝐿 are considered such that 𝑇 (𝐿) = 𝐿, 𝑇 ′ (𝐿) = 1, 𝑇 ′′ (𝐿) > 0. That
deduction fails, if 𝑇 ′′ (𝐿) = 0. One of examples of such a transfer function
is considered here.
This chapter deals with transfer function 𝑇 = sin. This function is often
used in various applications, so, I think, it deserves a special chapter.
Iterates of sin had been considered since century 19, but the rough approximations had been suggested only in 2011 [77, 78]. Then in 2014,
the efficient approximation had been reported in the Far East Journalof Mathematical Science [92]. Below, I retell the key ideas of that publication.
I hope, the Reader can plot the complex maps of sin and arcsin. I
recommend that Reader does this as an exercise. And also, the explicit
plots of these functions. After to watch the pictures mentioned, one canunderstand the sense of superfunction of sin, I call it SuSin. SuSun issolution of(︀the transfer)︀ equation, I repeat is once again,sin SuSin(𝑧) = SuSin(𝑧 +1)
Explicit plot of function SuSin of real argument is shown in figure 12.1.
In the next section, I describe the construction of this function.
𝑦
𝜋
2

𝑦=

√︀
3/𝑥

1

𝑦 = SuSin(𝑥)
0
0

1
2
3
4
5
6
7
8
9
https://mizugadro.mydns.jp/t/index.php/File:Susinplot.jpg
Fig. 12.1:

𝑦 = SuSin(𝑥) by (12.8) and its asymptotics (12.3)
148

𝑥

1

Super sin

Superfunction of sin, called SuSin, is shown in figure 12.1. It is solutionof the transfer equation with sin as transfer function; I repeat it once
again:(︁)︁𝐹 (𝑧 +1) = sin 𝐹 (𝑧)(12.1)In this section, the special solution 𝐹 = SuSin is constructed with thefollowing properties:

SuSun(0) = 𝜋/2
√︂ (︂
(︂
)︂)︂
3
ln(𝑧)
SuSun(𝑧) =1+𝑂𝑧𝑧(12.2)(12.3)The leasing term of the right hand side of (12.3) can be guessed replacing 𝐹 (𝑧 + 1) to 𝐹 (𝑧) + 𝐹 ′ (𝑧) in the left hand side of equation (12.1)
and solving the resulting differential equation. However, there is certain
residual at the substitution of such an approximation into (12.1); andthis residual indicates the form of the next term in expansion (12.3).

ln(𝑧)appears as the effective small parameter of the ex𝑧
pansion. The residual at the substitution of representation (12.3) into
the transfer equation (12.1) helps to guess a form of the higher term of
the expansion, and so on. After few such steps, I guess and verify thefollowing form of the approximation for the superfunction 𝐹 of sin:)︂(︂ln(𝑧)𝑀 +1
𝐹 (𝑧) = 𝐹𝑀 (𝑧) + 𝑂(12.4)𝑧 𝑀 +3/2

Expression

where

)︃
√︂ (︃
𝑀
∑︁
3
3 ln(𝑧)
𝐹𝑀 (𝑧) =1−+
𝑃𝑚 (ln(𝑧)) 𝑧 −𝑚
𝑧
10 𝑧
𝑚=2

𝑃𝑀 (𝑧) =𝑚∑︁

𝑎𝑚,𝑛 (−𝑧)𝑚(12.5)(12.6)

𝑛=0

and coefficients 𝑎 are constants. These constants can be calculated with
the Mathematica code below:

149

P[m_, L_] := Sum[a[m, n] L^n, {n, 0, m}]
F[z_] = Sqrt[3/z] ( 1 + Sum[P[m, Log[z]]/z^m, {m, 1, M}])M = 9; a[1, 0] = 0;F1x = F[1 + 1/x];
Ftx = Sin[F[1/x]];s[1] = Series[(F1x - Ftx)/Sqrt[x], {x, 0, 2}]t[1] = Extract[Solve [Coefficient[s[1], x^2] == 0, {a[1, 1]}], 1]A[1, 1] = ReplaceAll[a[1, 1], t[1]]su[1] = t[1]m=2;s[m]=Simplify[ReplaceAll[Series[(F1x-Ftx)/Sqrt[3 x],{x,0,m+1}], su[m-1]]]t[m] = Simplify[Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m+1)]]u[m] = Simplify[Collect[t[m], L]]v[m] = Table[Coefficient[u[m] L, L^(n+1)] == 0, {n, 0, m}]w[m] = Table[a[m, n], {n, 0, m}]
ad[m] = Extract[Solve[v[m], w[m]], 1]
su[m] = Join[su[m - 1], ad[m]]m=3;s[m]=Simplify[ReplaceAll[Series[(F1x-Ftx)/Sqrt[3 x],{x,0,m+1}],su[m-1]]]t[m] = Simplify[Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m+1)]]u[m] = Simplify[Collect[t[m], L]]v[m] = Table[Coefficient[u[m] L, L^(n+1)] == 0, {n, 0, m}]w[m] = Table[a[m, n], {n, 0, m}]
ad[m] = Extract[Solve[v[m], w[m]], 1]
su[m] = Join[su[m - 1], ad[m]]. . .

and so on. The resulting coefficients are shown in table 12.1.
For some positive integer 𝑀 , define function 𝐹 with(12.7)

𝐹 = lim arcsin𝑘 (𝐹𝑀 (𝑧 + 𝑘))𝑘→∞

The resulting 𝐹 does not depend on the number 𝑀 of terms taken into
account in the expansion (12.5). However, the larger, 𝑀 , the faster
Table 12.1: Table of coefficients 𝑎 in equation (12.6)

𝑎1,0
79
700
411
3500
1606257
10780000
140345627
700700000
137678711441
490490000000
25317035192599
62537475000000

3
10
9
50
1941
7000
7227
17500
70079931
107800000
73645237007000
8462569406199
4904900000000

𝑎1,2

𝑎1,3

𝑎1,4

𝑎1,5

𝑎1,6

𝑎1,7

27
200
27
125
1683
4000
566973700000305491257
196000000
32174780481
10780000000

𝑎2,3

𝑎2,4

𝑎2,5

𝑎2,6

𝑎2,7

27
400
1917
10000
98739
2000004155111
3500000
5367503637
1960000000

𝑎3,4

𝑎3,5

𝑎3,6

𝑎3,7

567
16000
7533
50000
7963111600000
407711313
280000000

𝑎4,5

𝑎4,6

𝑎4,7

15309
8000002218347
20000000
181900809
400000000

𝑎5,6

𝑎5,7

16839916000000
196028125000000

938223160000000

150

𝑎6,7

−0.
2

𝑦

𝑣=

8

𝑣=

0

cut

𝑣 = −1𝑣 =1

𝑢=1

𝑢 = 0.62

𝑢 = 0.7

4

𝑣 =0

.1
0
−

𝑢 = 0.5

6

−2

𝑣=

−4

𝑣=

−6

0 .2

−8

−10

0 .1

−8
−6
−4
−2
0
2
4
6
8
https://mizugadro.mydns.jp/t/index.php/File:Susinmap.jpg
Fig. 12.2:𝑥𝑢+i𝑣 = SuSin(𝑥+i𝑦) by (12.8)

is convergence of the limit in (12.7). This gives efficient algorithm forevaluation of superfunction 𝐹 for the transfer function sin.
For superfunction, declared in the beginning of the section, id est, satisfying equation (12.2), define

SuSin(𝑧) = 𝐹 (𝑧 +𝑥1 )(12.8)

where 𝑥1 ≈ 1.4304553465288 is solution of equation

𝐹 (1+𝑥1 ) = 1(12.9)

Note that the required value at zero is achieved, because SuSin(0) =arcsin(SuSin(1)) = arcsin(1) = 𝜋/2.
Function SuSin is shown as explicit plot and as complex map in figures
12.1 and 12.2. Function SuSin has sqrt-type singularity as zero, and it151

has cut line along the negative part of the real axis. In the rest of thecomplex plane, SuSin is holomorphic. At infinity, SuSin slowly decays
to zero, according to its asymptotic (12.3). This asymptotic is shown infigure 12.1 with thin line.
In this Book, SuSin is first example of superfunction, for which thetransfer function shows growth slower than linear. In this case, forpositive values of the argument (id est, larger than the fixed point), thesuperfunction is monotonically decreasing, as it is seen in Figure 12.1.
One could expect the inverse function (Abel sin) to decrease, at least forsome moderate positive values of the argument. This inverse function isconsidered in the next section.

AuSin

𝑦

https://mizugadro.mydns.jp/t/index.php/File:Ausinplot1.jpg

2Here, the inverse function of super sin is called AuSin; its explicit
plot is shown in figure (12.3). In 3
this chapter, I describe evaluationof this function.
For the sinusoidal transfer func- 2
tion, the Abel function 𝐺 = 𝐹 −1
satisfies the Abel equation

𝐺(sin(𝑥)) = 𝐺(𝑧)+1 (12.10)

1From the properties of SuSin, theproperties of the inverse function 00
can be guessed. In particular, forlarge values of the argument, theasymptotic below should hold:𝐺(𝑧) =

𝜋/2 2
Fig. 12.3: 𝑦 = AuSin(𝑥)2+ 𝑂(ln(𝑧))
𝑧2

1

𝑥

(12.11)

Abel sin can be constructed in analogy with super sin. The solution 𝐺of the Abel equation (12.10) with the asymptotic representation (12.11)
can be constructed, inverting function 𝐹 by (12.7), id est, 𝐺 = 𝐹 −1 .
Then, the constant 𝑥1 should be added,

AuSin(𝑧) = 𝐺(𝑧) + 𝑥1
152(12.12)

in order to satisfy relation

AuSin(𝜋/2) = 0(12.13)First, construct the approximation of function 𝐺; let𝑀∑︁
3 5
𝐺𝑀 (𝑧) = 2 + ln(𝑧)+𝑐𝑚 𝑧 2𝑚𝑧 6
𝑚=1(12.14)

Subsitution of 𝑔(𝑧) = 𝐺𝑀 + 𝑂(𝑧 2𝑀 +2 ) into the Abel equation (12.10)
gives the coefficients 𝑐. In particular,

79
1050
29
𝑐2 =
2625
91543
𝑐3 =
36382500
18222899
𝑐4 =28378350000

𝑐1 =(12.15)(12.16)(12.17)(12.18)

Then, for some fixed 𝑀 , function 𝐺 can be expressed as limit

𝐺(𝑧) = lim 𝐺𝑀 (sin𝑘 (𝑧)) − 𝑘𝑘→∞(12.19)and AuSin = SuSin−1 appears as

AuSin(𝑧) = 𝐺(𝑧) − 𝐺(𝜋/2)(12.20)Term 𝐺(𝜋/2) ≈ 2.089622719729524 in equation (12.20) provides condition (12.13).
Complex map of abelsinus AuSin by (12.20) is shown in fgure 12.4. As
for other real-holomorphic functions, this map is symmetric with respect
to reflection from the real axis, id est, with respect to flipping upsidedown. In addition, the map is symmetric with repeat to reflections fromthe axis 𝑥 = 𝜋/2; the first evaluation of function sin in formula (12.19)
provides this symmetry.
For the central part of figure 12.4, the limit (12.19) converges and defines the holomorphic function in a pretty regular way. At lateral partsof figure 12.4, the lines of level of the real part and those of the imaginary part form the fractal-like structures. There, the AuSin cannot be
considered as inverse function of SuSin. Actually, the range of validityof the inverse function is even narrower.
153

−2

𝑦

−1
=
𝑢

𝑢=

1

𝑢=2
𝑢=1

𝑢 = −0.5

𝑢=

−1

𝑣=

0

1

0

𝑣 =0

𝑣 = 0.2𝑣 =0𝑣 = −0.2

𝑢=

𝑣 =1

𝑣 =3
𝑣 =2

0

−1

0

𝜋/2

2

https://mizugadro.mydns.jp/t/index.php/File:Ausinmap52r5.jpg
Fig. 12.4:

3𝑥𝑢+i𝑣 = AuSin(𝑥+ i𝑦) by formula (12.20)

Range of validity of relation(︁)︁
SuSin AuSin(𝑧) = 𝑧(12.21)is shown in figure 12.5, this range is shaded with rectangular grid.

154

𝑦

1

0

−1

0

1

𝜋/2

2

3𝑥

https://mizugadro.mydns.jp/t/index.php/File:Ausinsusinmapt50.jpg
Fig. 12.5: 𝑢+i𝑣 = ℎ(𝑥+i𝑦) and lines ℑ(AuSin(𝑥+i𝑦) = 0

Technically, the shading in figure 12.5 is built up as the complex mapof function(︁)︁ℎ(𝑧) = SuSin AuSin(𝑧)(12.22)

In addition, in figure 12.5, lines of level ℑ(AuSin(𝑥 + i𝑦)) are shown;
they are borrowed from figure 12.4. These lines bound the range ofvalidity of relation ℎ(𝑧) = 𝑧 .
155

The readers are cordially invited to download the generator of the figure
and modify it, to plot the map of the agreement function(︂|ℎ(𝑧) − 𝑧|
𝐴(𝑧) = − lg|ℎ(𝑧)| + |𝑧|)︂(12.23)and to verify, that in the shaded part, relation AuSin = SuSin−1 holdswith at least 14 decimal digits. At the right hand side of the figure 12.5,
relation AuSin = SuSin−1 is not valid; AuSin(𝑥+i𝑦) is symmetric withrespect to line 𝑥 = 𝜋/2; so, it cannot distinguish values for the righthand side in figure 12.3 from those at the left hand side.
For the efficient (fast and precise) numerical implementation of AuSin, various expansions canbe used. The Taylor expansion at 𝜋/2 can be
written as follows:

AuSin

(︁ 𝜋
2

)︁

+𝑡 =

∞
∑︁

𝑏𝑛 𝑡

(12.24)2𝑛𝑛=1Coefficients 𝑏 in the expansion (12.24) are evaluated with representation of AuSin through function 𝐺 by (12.14), (12.19), (12.20). Approximations of first nine of these coefficients are shownin Table 12.2.

Table 12.2

𝑛𝑏𝑛
1 2.29163807440958
2
3
4
5
6

1.96043852439688
1.07862851256147
0.59622997993395
0.28333997139829
0.14193261194548

7 0.06423734271234
8 0.03026687705508
9 0.01351721250427

Series (12.24) converges at |𝑡| < 𝜋/2. It is sufficient to take few terms of this expansion inorder to reproduce the most of map in figure 12.4. Readers are invited
to plot this map (or, at least, to look for it in the Appendix, figure 22.2
at page 300).

Taking into account some tens of coefficients in series (12.24), the numerical approximations of AuSin provides the precision at least not worse,
than the precision of the original representation through the asymptotic
formulas (12.14), (12.19), (12.20). However the original representation isstill necessary to calculate the coefficients of the secondary expansions,and, in particular, those of the Taylor expansion (12.24). Optimisationof such representations may have sense before to include them to some
software as built-in functions, while the each microsecond at the evaluation is important for the resulting efficiency at the multiple evaluations.
However, even with the primitive approximations described above, thefunctions are evaluated with approximately 14 decimal digits and allow
to plot complex maps in real time. This indicates the good efficiency of
the representations suggested.
156

0

𝑛=

−0
.2
𝑛=

𝑛 = −20𝑛 = −10𝑛 = −5𝑛 = −3𝑛 = −2𝑛=−1

1𝑛 = −100𝑦𝑛 = 0..21𝑛= 0
𝑛 = 0.4

1𝑛=𝑛 = 23𝑛=𝑛= 5𝑛 = 10𝑛 = 20𝑛 = 100

00

1

Fig. 12.6:

3

𝜋/2

2

https://mizugadro.mydns.jp/t/index.php/File:Sinplo2t100.jpg

3 𝑥

𝑦 = sin𝑛 (𝑥) by (12.25) versus 𝑥 for various 𝑛

Iterated sin

With functions SuSin and AuSin = SuSin−1 , defined above, the iteratesof sin can be expressed as follows:)︁sin (𝑧) = SuSin 𝑛 + AuSin(𝑧)𝑛(︁(12.25)This formula looks pretty similar to representations of iterates of any
other function with determined superfunction and the corresponding
Abel function. In this representation, number 𝑛 of iterates has no need
to be integer; it can be real and even complex. For real values of number
𝑛 of iterates, sin𝑛 (𝑥) is plotted versus 𝑥 in figure 12.6.
For positive number 𝑛 of iterate, graphics of 𝑦 = sin𝑛 (𝑥) are symmetricwith respect to line 𝑥 = 𝜋/2. The larger is 𝑛, the closer the curve
approaches the abscissa axis.
For negative 𝑛, the graphic reaches the branch point at 𝑦 = 𝜋/2 andcannot be continued be continued above, as the iterates get complex
values. A usually, the 0th iterate corresponds to the identity function,
sin0 (𝑥) = 𝑥, and this relation holds while 𝑥 < 𝜋/2.
Through the iterates (12.25) of sin, the SuSin can be expressed as follows:

SuSin(𝑧) = sin𝑧 (𝜋/2)(12.26)

From the point of view of computation, representation (12.26) does nothave much sense. Anyway, for the evaluation of the right hand side of(12.26), the approximations of SuSin should be used. However, such a
representation may have sense at the building-in of superfunctions (andnon-integer iterates) into the programming languages.
157

4

Application

In this section, I discuss applications of multiple iterates of sin. However,I assume, that the Readers can freely use SuSin and AuSin, describedabove, to plot new graphics. I load the complex double implementationsof these functions as
https://mizugadro.mydns.jp/t/index.php/Susin.cin andhttps://mizugadro.mydns.jp/t/index.php/Ausin.cin

Beautiful figures can be plotted playing with SuSin and AuSin.
For the physical applications, the real number of iterate is simpler tointerpret, than the complex iterate. Keeping in mind the application, I
suggest the example of parameterisation of the shape of the sled runnerwith high iterate of sin. Figure 12.7 shows(12.27)𝑦 = sin𝑛 (𝜋/2)−sin𝑛 (𝑥)plotted over the photo of the sled (with a boy in it), taken from Wikimedia Common [9].
The curve, that overlaps wight he sled runner, is the 100th iterate of sin;
the number of iterate 𝑛 = 100 is the only adjusted parameter used forthe fitting. However, the photo is shifted, scaled and rotated, in order tohave the tip at 𝑥 = 0 and the last support of the sled runner at 𝑥 = 𝜋/2,
𝑦 = 0; in this point the sled runner is horizontal, this is provided by theslight rotation of the photo (for one degree).

𝑦
0.1
0 0

1

𝜋/2

https://mizugadro.mydns.jp/t/index.php/File:Boyt100.jpg
Fig. 12.7: Boy at the sledge and 𝑦 = sin158𝑛(𝜋/2)−sin𝑛 (𝑥) for 𝑛 = 100

2 𝑥

𝑥

1

(13.1)

−2
−3

𝑞 =0

I assume, that 𝑞 is positive real parameter. For
various values of 𝑞 , the explicit plot of this function is shown in figure 13.1. This is simple, but
still non-trivial case of the transfer function 𝑇 ,for which

−1

0.5
𝑞=

This chapter represents the last (for this book)example of exotic iteration, and deal with thespecific polynomial

𝑞 =1

−1 00

−4

https://mizugadro.mydns.jp/t/index.php/File:Nemplot.jpg

1

3
2

Nemtsov function

𝑇 (𝑧) = Nem𝑞 (𝑧) = 𝑧 + 𝑧 3 + 𝑞𝑧 4

𝑞 =0

𝑞 =2

Chapter 13

4

𝑞 =2

𝑦

Fig. 13.1: 𝑦 = Nem𝑞 (𝑥)𝑇 (𝐿) = 𝐿 , 𝑇 ′ (𝐿) = 1 , 𝑇 ′′ (𝐿) = 0 , 𝑇 ′′′ (𝐿) > 0 , 𝑇 ′′′′ (𝐿) > 0 (13.2)
Note, that for 𝑇 = sin, considered in the previous section, 𝐿 = 0 and𝑇 ′′′′ (𝐿) = 0; function sin is antisymmetric, 𝑇 (−𝑧) = −𝑇 (𝑧), and, in thissense, is simpler, than 𝑇 = Nem𝑞 .
The analysis of transfer function 13.1) is my last attempt (up to year2020) to write an elementary function, for which I would not be able
construct superfunction, abelfunction and the non-integer iterates. I
found, that I need a special name for function by (13.1). That happened
2015.01.28; the same day, murder of Boris Nemtsov had been reported 1 ;so I pick up the first three letters of his last name to denote the function.
In this Book, I do not follow the historical timeline of events, so, I put
chapter about the Nemtsov function here, in order to have the exotic iterations in one bunch. For real argument, graphics of the Nemtsov function for various values of parameter 𝑞 are shown in figure 13.1. Complex
maps of function Nem𝑞 and its inverse function ArqNem𝑞 = Nem−1
𝑞 areshown in figure 13.2 for 𝑞 = 0, 𝑞 = 1 and 𝑞 = 2. Function ArqNem is
described in the following section.
1 https://mizugadro.mydns.jp/t/index.php/Putin_killed_Nemtsov
159https://mizugadro.mydns.jp/t/index.php/File:Nem0map.jpg

0.8

cut

10

5

−1
𝑥 −1

10
−1

0

1

𝑥

0.3
𝑣=

−1
𝑥 −1

𝑣 =0

−
𝑣=

𝑢=
−0.5

.1

𝑣 =−
0.1

0.
6

cut

0 .2

𝑣 =0

𝑢=0

10
𝑣=
−2
−2

10

𝑣=

𝑢=1
𝑢=2
𝑢=5

𝑢=0

𝑣 =0

− 21

−1

−
𝑣=

𝑢 = −10 12
10 𝑦 0
=
𝑢
𝑣=
𝑣 =5
0.7
0
5
−
𝑣=
𝑢=
10 −𝑦0

.5

𝑢=0

0.5

6
0.

𝑣 =0

𝑦

𝑣=

0
−1

𝑣 =0
0

0
−1

0

𝑣 =0

𝑣=
1

𝑣=

10

0

𝑣=

𝑢=

𝑦

−0.5

https://mizugadro.mydns.jp/t/index.php/File:ArqNem2map.jpg

𝑣 = 0.4https://mizugadro.mydns.jp/t/index.php/File:Nem2map.jpg

𝑢𝑢 == 0.2
𝑢 = 0.3
0.4
𝑢=
0.5

10

cut

1

𝑣=

𝑣=

0

0
−1

−1

0 .3

0.4

𝑣 =−
0.1

𝑣=

−2
−2

0 .2
𝑣=
1
𝑣 = 0.

𝑣 =0

− 12

𝑣=

𝑣=

𝑣=

10

𝑣=

−𝑦0

−1

−5
𝑣 = −10𝑣=𝑥𝑢𝑢 == 0.2
𝑢 = 0 .3
𝑢 = 0.4
0 .5
𝑢=
0.
6

𝑢=1
𝑢=2
𝑢=5
𝑢=0

𝑢=

𝑣 =0

𝑢=0

𝑢=

−1

10 𝑦0
=
𝑢
𝑣 =5
5 0
𝑣 =−

0 .6

2

7

𝑣 =0

−0.
2

0.5

0.

0

𝑦

𝑢 = −10 1

=0

0

𝑣=

𝑣

10

𝑣=

10

−0.5

1

https://mizugadro.mydns.jp/t/index.php/File:ArqNem1map.jpg

𝑣=

−
𝑣=

1

−1

−1
𝑥 −1

1

𝑢=0

𝑣=

𝑣=

𝑣=

−𝑦0
− 12

https://mizugadro.mydns.jp/t/index.php/File:Nem1map.jpg

𝑦𝑢=0
𝑢 = 0..1
𝑢= 2
𝑢 = 0 .3
𝑢=0 .4
𝑢 = 0 .5
0 .6
𝑢=
0.7

𝑣 = −0.

−5

𝑢 = −5𝑢 = −10

0

4
0.
0 .3
𝑣=
.2
𝑣 =0
𝑣 = 0.1𝑣 =0

𝑣 = 0.5

−1

cut

cut

−2
−2

1

𝑢=0

𝑣=
𝑣 = −1
0

𝑣=

cut

𝑢=5

𝑢=1
𝑢=2

𝑢=0

𝑢 = −1

𝑢 = −5

𝑢=0

−1

𝑣=

1

𝑣=

0

1
1.

𝑣 =0

5

𝑣=

0

𝑣=

𝑣=

𝑢=0

𝑣=

1
2
𝑦0

1

𝑣 = 0.7
𝑣=
0.6
𝑣=
0 .5

𝑦

𝑣 = 0.8
0.9

𝑢 = 10𝑢=5

https://mizugadro.mydns.jp/t/index.php/File:ArqNem0map.jpg

𝑢=0

𝑦

0

0.5

6
0.
𝑥

Fig. 13.2: Maps 𝑢+i𝑣 = Nem𝑞 (𝑥+i𝑦) for 𝑞 = 0, 1, 2, left column, and maps 𝑢+i𝑣 =ArqNem𝑞 (𝑥+i𝑦), right column, for the same 𝑞

160https://mizugadro.mydns.jp/t/index.php/File:Arqnemplot.jpg

𝑦 = ArqNem𝑞 (𝑥)
𝑞 =0
𝑞 =1
𝑞 =2
𝑞 =4
𝑞 = 10

1
0.5
0.5

1

1.5

2

2.5

3

𝑥

Fig. 13.3: 𝑦 = ArqNem𝑞 (𝑥) for 𝑞 = 0, 1, 2

1

ArqNemFor the efficient evaluation of iterates of a function, we need both, its
superfunction and the abelfunction. For implementations of these functions, we need both the transfer function 𝑇 = Nem𝑞 and its inverse
ArqNem𝑞 = 𝑇 −1 . For real values of the argument, plot 𝑦 = ArqNem𝑞 (𝑥)for 𝑞 = 0, 𝑞 = 1, 𝑞 = 2, 𝑞 = 4, 𝑞 = 10 is shown in figure 13.3. Inversionof the Nemtsov function happens to be not trivial, and its description
deserves the special section.
The Nemtsov function Nem𝑞 by (13.1) is the 4th order polynomial; for𝑞 > 0, equation

Nem𝑞 (𝑥) = 𝑧(13.3)

at given 𝑧 has four solutions 𝑥. Any of original solutions (suggested, forexample, by Mathematica routine Solve) happen to be ugly at the complex map, and even worse at evaluation of abelfunction for the Nemtsov
function. The reader can plot the complex maps for the four rootsof equation (13.3) and see the root of my discontent with them. But
these solutions can be used to construct the inverse function shown theright hand column of figure 13.2. I have implemented several combinations of the “primary” solutions, giving special name to each resulting
inverse function: ArcNem𝑞 , ArkNem𝑞 , ArqNem𝑞 . The third of them,
ArqNem𝑞 , happened to be satisfactory; so, I use it as Nem−1 . The cut
lines of function ArqNem𝑞 go from −∞ to zero, and then to each of the
branch points, seen at the maps in the right column of Figure 13.2.
For the positive values of the argument, curves in Fig. 13.3 refer to anyof these three functions, ArcNem𝑞 (𝑥) = ArkNem𝑞 (𝑥) = ArqNem𝑞 (𝑥) .
Below I describe the only last one.
161

𝑞 =4
𝑞 =5

Assume, some 𝑞 is given. We need to find solutions of equation Nem𝑞′ (𝑧) = 0. This solu0.2
tion can be expressed in Mathematica languagewith code

T[z_]=z+z^3+q z^4
s = Solve[T’[z]==0,z]ReplaceAll[z,Extract[s,1]]

0.1

The output indicates, how to program function
NemBran, shown in figure 13.4:

z_type nembra(DB q){ z_type Q,v,V;
Q=q*q;
v=-1.-8.*Q+4.*sqrt(Q+4.*Q*Q);
V=pow(v,1./3.);
return (.25/q)*(-1.+1./V+V); }𝑞 = 10𝑞 = 20𝑞 = 30𝑞 = 100
𝑞 = 1000
𝑞 = 10000

0

0

0.05

0.1

https://mizugadro.mydns.jp/t/index.php/File:Nembrant.jpg

For handling of cuts of function ArqNem𝑞 , the 𝑦𝑞 = 0.2
𝑞 = 0.4
branch points should be calculated. These 𝑞 = 0
𝑞 = 0.6
𝑞 = 0.8
branch points are shown in figure 13.4. It refers
𝑞 =1
to function NemBran, that expresses the upper
branch point of function ArqNem𝑞 as function 0.3
𝑞 =2
of 𝑞 . Below I describe, how to evaluate this𝑞 =3function.

𝑥Fig. 13.4: Parametric plot:

𝑥+i𝑦 = NemBran(𝑞)z_type NemBran(DB q){ z_type z,zz=z*z;
z=nembra(q); return z*(1.+zz*(1.+q*z)); }Here DB denotes <double> and z_type denotes complex<double> ;NemBran appears as combination of functions Nem and nembra.
Assume, some 𝑞 > 0 is given; let 𝑥0 +i𝑦0 = NemBran(𝑞). For function
ArqNem𝑞 , I draw the cut lines form −∞ to zero, and from zero to point(𝑥0 , 𝑦0 ) and to point (𝑥0 , −𝑦0 ). As the cut lines are specified, it is easy toprogram evaluation of ArqNem𝑞 , picking up the corresponding branchof the solution. The C++ code is shown in Table 13.1. One can extract
the code also from the URL marked in figure 13.4. The readers areinvited to check numerically relations(︀)︀
Nem𝑞 ArqNem𝑞 (𝑧) = 𝑧(︀)︀ArcNem𝑞 Nem𝑞 (𝑧) = 𝑧
and investigate the range of validity of each of these equation.
162(13.4)(13.5)

Table 13.1:

C++ implementation of function

ArqNem𝑞 .

Values of𝑞 and corre-

sponding 𝑥0 , 𝑦0 should be already calculated and stored in global variables Q, tr, ti

z_type arnemU(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);z_type R=-I*sqrt(-r);
z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=I*sqrt(-h);
z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H);z_type G=I*sqrt(-g);
return - 0.25/q - 0.5*H + 0.5*G ;}z_type arnemD(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);z_type R=I*sqrt(-r);
z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=-I*sqrt(-h);
z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H);z_type G=-I*sqrt(-g);
return - 0.25/q - 0.5*H + 0.5*G ;}z_type arnemR(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);z_type R=sqrt(r); z_type s=27.*a + 3.*R;
z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=sqrt(h);z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H);z_type G=sqrt(g);return - 0.25/q - 0.5*H + 0.5*G ;}z_type arnemL(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);z_type R=-sqrt(r);z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=sqrt(h);z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H);z_type G=sqrt(g);return - 0.25/q - 0.5*H + 0.5*G ;}z_type arqnem(z_type z){ DB x,y; x=Re(z);y=Im(z);if( y>ti || (x<0 && y>=0)) return arnemU(z);//if(y<0) return conj(arnemU(conj(z)));if(y<-ti || (x<0 && y<=0)) return arnemD(z);if(x*ti>fabs(y)*tr) return arnemR(z);return arnemL(z);}

163

8
7
6
5
4

For the Nemtsov function Nem𝑞 , the superfunction is solution 𝐹 of the transfer equation

3(13.6)

2

In analogy with approach of the previous
chapter, I look for solution 𝐹 with the certain
asymptotic behaviour,1

𝐹 (𝑧 +1) = Nem𝑞 (𝐹 (𝑧))

−1

0

1

https://mizugadro.mydns.jp/t/index.php/File:Sunemplo4t.jpg

Functions Nem𝑞 and ArqNem𝑞 =are defined, implemented, tested and described above; they can be used for construction of superfunction SuNem𝑞 , abelfunction
AuNem𝑞 = SuNem−1
𝑞 and corresponding noninteger iterates of the Nemtsov function. I
begin with function SuNem𝑞 . It is shown infigure 13.5 for various values of 𝑞 . In thissection, I describe, how is it constructed.

𝑞 =0

9

Nem−1
𝑞

𝑞 =8

SuNem

𝑞 =2
𝑞 = 0.5 𝑞 = 1

2

𝑦

𝑥

Fig. 13.5: 𝑦 = SuNem𝑞 (𝑥)(︂(︁ ln(−𝑧) )︁)︂
𝑞
1
1− √
+𝑂
𝐹𝑞 (𝑧) = √
𝑧
−2𝑧
−2𝑧

(13.7)In order to construct the computationally-efficient asymptotic approximation of superfunction 𝐹𝑞 , define set of polynomials
IntegerPart[𝑚/2]𝑃𝑚 (𝑧) =∑︁

(13.8)

𝑎𝑚,𝑛 𝑧 𝑛𝑛=0where 𝑎 are constant coefficients. Then, I set(︃
𝐹𝑞,𝑀 (𝑧) = 𝜀 1 − 𝑞𝜀 +

𝑀
∑︁

)︃
𝑃 (ln(−𝑧))𝜀𝑚(13.9)

𝑚=2

where

𝜀=√

1
−2𝑧

(13.10)I substitute the approximation (13.9) into the transfer equation (13.6).
The asymptotic analysis of the residual (id est, its asymptotic minimisation) determines coefficients 𝑎. The asymptotic representation(13.8),(13.9),(13.10) should approximate superfunction 𝐹 at least forlarge negative values of the argument.
164

Table 13.2: Computation of coefficients 𝑎 in equation (13.8)

T[z_] = z + z^3 + q z^4
P[m_, L_] := Sum[a[m, n] L^n, {n, 0, IntegerPart[m/2]}]
A[1, 0] = -q; A[1, 1] = 0;a[2, 0] = 0; A[2, 0] = 0;
F[m_,z_]:=1/(-2z)^(1/2)(1-q/(-2z)^(1/2)+
Sum[P[n,Log[-z]]/(-2z)^(n/2),{n,2,m}])
m = 2;
s[m] =Numerator[Normal[Series[(T[F[m,-1/x^2]]-F[m,-1/x^2+1])2^((m+1)/2)/x^(m+2),{x,0,1}]]]t[m] = Numerator[Coefficient[Normal[s[m]], x] ]sub[m] = Extract[Solve[t[m] == 0, a[m, 1]], 1]
SUB[m] = Simplify[sub[m]]f[m, z_] = ReplaceAll[F[m, z], SUB[m]]m = 3s[m] =Simplify[ReplaceAll[Series[(T[F[m,-1/x^2]]-F[m,-1/x^2+1])2^((m+3)/2)/x^(m+3),{x,0,0}],SUB[m-1]]];t[m] = ReplaceAll[Normal[s[m]], Log[x] -> L];u[m] = Table[Coefficient[t[m] L, L^n]==0, {n,1,1+IntegerPart[m/2]}];tab[m] = Table[a[m, n], {n, 0, IntegerPart[m/2]}];sub[m] = Extract[Solve[u[m], tab[m]], 1]
SUB[m] = Join[SUB[m-1], sub[m]];(* and so on for

m=4,

m=5, etc. *)The original (and non-trivial) part of this research is guessing of representation (13.8),(13.9),(13.10). This representation is a little bit more
complicated, that the similar representation (12.4), (12.5), (12.6) for
function SuSin, described in the precious section.
Once representation (13.8),(13.9),(13.10) is written out, the following
construction is straightforward. This analysis can be performed with
the Mathematica code shown in Table 13.2.
Coefficients 𝑎 are chosen in such a way, that(︀)︀
𝐹𝑞,𝑀 (𝑧) = 𝐹𝑞 (𝑧) + 𝑂 𝜀𝑀 +1(13.11)

Tens of coefficients 𝑎 in equation (13.8) can be computed in such a way.
The first coefficients are:𝑎2,0 = 0 ,𝑎2,1 = 41 (3 + 2𝑞 2 )𝑎3,0 = 𝑞 + 3𝑞 3 ,𝑎3,1 = −(3𝑞)/2 − 𝑞 33𝑎4,0 = 81 (5−4𝑞 2 −44𝑞 4 ), 𝑎4,1 = 18 (−9−12𝑞 2 −4𝑞 4 ), 𝑎4,2 = 32
(9+12𝑞 2 +4𝑞 4 )1
𝑎5,0 = 12(−39𝑞−104𝑞 3−4𝑞 5 ), 𝑎5,1 = 74 (3𝑞+8𝑞 3+4𝑞 5 ), 𝑎5,2 = − 9𝑞4 −3𝑞 3−𝑞 5165

Coefficients 𝑎 happen to be polynomials with respect to parameter 𝑞 .
For the following numerical implementation of superfunction, they are
expressed through the Horner rule. The C++ implementation is loaded
together with generators of figures of this chapter.
Assume given number 𝑀 of terms of sum in the asymptotic expansion(13.9). Then, the superfunction 𝐹𝑞 can be defined as limit

𝐹𝑞 (𝑧) = lim Nem𝑞𝑛
𝑛→∞

(︁

)︁
𝐹𝑞,𝑀 (𝑧 − 𝑛)(13.12)

I remind, the upper index after the name of the function indicates thenumber of its iterate. The limit does not depend on the number 𝑀 .
However, for large 𝑀 , the limit converges faster. For 𝑞 of order ofunity, and argument 𝑧 of order of unity, with 𝑀 = 30, it is sufficient tomake 𝑛 = 20 iterates in order to approximate limit in equation (13.12)
with 14 decimal digits. That greatly exceeds the precision, required toplot the camera-ready copies of all the figures presented. However, theextra digits help to reveal faults of the representation of function, if any
mistake takes place.
The transfer equation has translational invariance. If some 𝑧 → 𝐹 (𝑧)is the solution, then, for a constant 𝐶 , function 𝑧 → 𝐹 (𝑧 +𝐶) is also
solution, id est, also superfunction of the same transfer function. In
order to make figures more beautiful it is convenient, that at zero, thesuperfunction has value unity. For this reason, I define superfunction
SuNem𝑞 as superfunction 𝐹 with displaced argument:

SuNem𝑞 (𝑧) = 𝐹𝑞 (𝑥1 +𝑧)(13.13)where 𝑥1 = 𝑥1 (𝑞) is real solution of equation

𝐹𝑞 (𝑥1 ) = 1(13.14)

SuNem𝑞 (0) = 1(13.15)This provides condition

This condition is not so important to evaluate iterates of the Nemtsov
function, but it helps to compare different superfunctions. Many superfunctions, described in this Book, have value unity at zero.
Definition (13.13) of function SuNem is used to generate the explicit plot
in figure 13.5 and also the complex maps of SuNem𝑞 for 𝑞 = 0, 𝑞 = 1 and𝑞 = 2 in figure 13.6. These maps are symmetric with respect to reflection
from the real axis, so, the only upper half of the complex plane is shown.
166https://mizugadro.mydns.jp/t/index.php/File:Sunem0ht.jpg

3

0 .1

0 .1

0 𝑣 =0−5−4

𝑢=0

𝑢 = 0 .4

1

𝑞 =0

−3

𝑢=
0.6
𝑢=
𝑢 = 0.7
0.
𝑢= 8
1

2

.5

𝑣=

𝑣 = 0 .4𝑢=𝑣 = 0.2

4

.3

0 .3

.2

𝑢=0

𝑣=

𝑢=0

𝑦

−2

−1

0

1

2

3

huge values

4𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sunem1ht.jpg

0 .1

3

0.1

𝑞 =1

0 .5
−2

𝑢=
𝑢 = 0.6
𝑢 = 0.
7
𝑢 = 0.81𝑣 = 0−3

−4

𝑢=

𝑢=0

0
−5

.4

𝑢=

0.3

2
1

𝑣 = 0 .3

𝑢=

𝑣=

𝑣=

4

0 .2

0.2

𝑢=

𝑦

−1

0

1

2

3

huge values

4𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sunem2ht.jpg

0.2

𝑦

𝑣=

𝑢=

0 .2

0.1

3

0.1

𝑢=

𝑣=

𝑣=

4

𝑞 =2

0 .3

−4

.4

𝑣 = 0−3−2𝑢=0
𝑢 = .5𝑢 0
𝑢 == 0 .6
𝑢 = 0..7
18

0
−5

𝑢=0

1

𝑢 = 0 .3

2

−1

0

1

2

Fig. 13.6: 𝑢+i𝑣 = SuNem𝑞 (𝑥+i𝑦), for 𝑞 = 0, 1, 2
167

3huge values

4𝑥Maps in figure 13.6 demonstrate the asymptotic behaviour of function
SuNem. In the most of the complex plane, the function slowly decays
to zero. This decay is determined by the leading term of the asymptotic
expansion (13.9). However, this asymptotic expansion is not valid in
vicinity of the origin of coordinate, nor in vicinity of the positive partof the real axis. In the half-strips 𝑥 > 2, 1 < |𝑦| < 2, the maps showcomplicated, oscillating behaviour of the function. For positive valuesof the argument, function SuNem𝑞 shows fast growth. This growth isseen both, in the explicit plot at the right hand side picture of figure13.5 and in the complex maps in figure 13.6.
For computation of iterates of the Nemtsov function Nem𝑞 , the Abelfunction AuNem𝑞 = SuNem−1
is also required. Function AuNem𝑞 is𝑞described in the next section.

3

AuNem

In this section, I construct abelfunction for the Nemtsov function Nem𝑞by (13.1). This abeldunction is inverse of the superfunction, id est,
AuNem𝑞 = SuNem−1
𝑞 .
First, consider inverse function of the superfunction 𝐹 by (13.7); let
𝐺𝑞 = 𝐺 = 𝐹 −1 . It can be expanded as follows:𝑞 1 (︀ 2 )︀𝑞 2 1 (︀ 2 )︀1
2𝑞 +3 log(𝑧) + +2𝑞 +3 log(2)G𝑞,𝑀 (𝑧) = − 2 + +
2𝑧
𝑧 2
2
4
𝑀
∑︁
𝑐𝑛 𝑧 𝑛(13.16)+𝑛=1

𝐺𝑞 (𝑧) = G𝑞,𝑀 (𝑧) + 𝑂(𝑧 𝑀 +1 )(13.17)This form can be obtained, inverting expansions (13.7), (13.9) for superfunction 𝐹𝑞 .
Coefficients 𝑐 depend on 𝑞 ; these coefficients can be computed eitherwith asymptotic analysis of equation

𝐺𝑞,𝑀 (𝐹𝑞,𝑀 (𝑥)) = 𝑧(13.18)or from the Abel equation

𝐺𝑞 (Nem𝑞 (𝑧)) = 𝐺𝑞 (𝑧) + 1(13.19)Coefficients 𝑐 in equation (13.16) can be calculated in Mathematica withthe code shown in Table 13.3.
168

Table 13.3: Mathematica code for calculation of coefficients 𝑐 in equation (13.16)

T[z_] = z + z^3 + q z^4
P[m_,L_]:= Sum[a[m, n] L^n, {n, 0, IntegerPart[m/2]}]
F[m_,z_]:=1/(-2z)^(1/2)(1-q/(-2z)^(1/2)+
Sum[P[n,Log[-z]]/(-2z)^(n/2),{n,2,m}])G[m_,x_]:=-1/(2x^2)+q/x+q^2/2+1/4(3+2q^2)Log[2]+1/2 (3+2q^2)Log[x]+Sum[c[n]x^n,{n,1,m}]
Series[ReplaceAll[F[3,h+G[3, z]], a[2,1]-> 1/4 (3+2 q^2)], {z,0,4}](*The line above is just test *)m=1;
sg[m]=Coefficient[Series[G[m+3,T[z]]-G[m+3,z]-1,{z,0,3}], z^(m+2)]
st[m]=Solve[sg[m] == 0, c[m]]su[m]=Extract[st[m], 1]
SU[m]= su[m];m= 2;
sf[m]=Series[ ReplaceAll[G[m+3,T[z]]-G[m+3,z]-1,SU[m-1]],{z,0,m+2}]
sg[m]=Simplify[Coefficient[sf[m] 2^m, z^(m+2)]]
st[m]= Solve[sg[m] == 0, c[m]]SU[m]= Join[SU[m - 1], su[m]]m = 3;
sf[m]=Series[ ReplaceAll[G[m+3,T[z]]-G[m+3,z]-1,SU[m-1]],{z,0,m+2}]
sg[m]=Simplify[Coefficient[sf[m] 2^m, z^(m+2)]]
st[m]= Solve[sg[m] == 0, c[m]]su[m]= Extract[st[m], 1]
SU[m]= Join[SU[m-1], su[m]](*and so on for m=4, m-5, etc... *)For an integer 𝑀 > 0, the abelfunction 𝐺𝑞 can be evaluated through theasymptotic approximation 𝐺𝑞,𝑀 as the limit)︁)︀𝐺𝑞 (𝑧) = lim G𝑞,𝑀 ArqNem𝑞 (𝑧) + 𝑛
(︁

𝑛

(︀

𝑛→∞

(13.20)Then, the function AuNem is expressed through function 𝐺 with addition of the constant,

AuNem𝑞 (𝑧) = 𝐺𝑞 (𝑧) − 𝐺𝑞 (1)(13.21)in such a way, that AuNem(1) = 0. The readers are invited to check
numerically the ranges of validity of relation(︀)︀
SuNem𝑞 AuNem𝑞 (𝑧) = 𝑧(︀)︀
AuNem𝑞 SuNem𝑞 (𝑧) = 𝑧(13.22)(13.23)and also estimate the residuals at the substitution of the numerical implementations into these relations.
169https://mizugadro.mydns.jp/t/index.php/File:Aunem0ht.jpg//

𝑢=

2.1

𝑢 = 2 .2

0.5

1

2

𝑣 =2

𝑢=

𝑢=

𝑢 = 0.5𝑣 = 1.7𝑣 = 1 .8

𝑦

cut

𝑣=
cut

2 .5
𝑢=

0
−1

−0.5

0

0

𝑢 = 2 .3

𝑢=

1

𝑦0

𝑣 =0 𝑥0.5

https://mizugadro.mydns.jp/t/index.php/File:Aunem1ht.jpg

𝑢=

𝑢=1

𝑦

1 .5

𝑢 = 1 .6

1 .8

cut

cut
0
−1

𝑢=

𝑢 = 1 .7
−0.5

0 𝑥0

0.5

0

𝑣=

𝑢=

1

𝑦0

𝑢 = 0.5𝑣=
𝑣 = 1 .4
1 .5

0.5

1 .4

𝑢=

𝑣 =0

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Aunem2ht.jpg

.8

1 .3

1.
𝑣=

1
𝑣=
𝑢

cut

cut
0
−1

.5
1
=

−0.5

0 𝑥0

0.5

Fig. 13.7: 𝑢+i𝑣 = AuNem𝑞 (𝑥+i𝑦) for 𝑞 = 0, 𝑞 = 1, 𝑞 = 2

170

.1
−0

𝑢 = 1 .4

𝑢 = 0.4
𝑢 = 0 .3
𝑢=0
𝑢 = .2
𝑢 = 0.1
0

𝑢=

𝑦0

1.3

𝑣=

0.5

2

1 .2
𝑢=

𝑢=0

𝑢=

𝑢=1

𝑦

𝑣 =0

𝑥

Complex maps 𝑢+i𝑣 = AuNem𝑞 (𝑥+i𝑦) for 𝑞 = 0, 𝑞 = 1 and 𝑞 = 2 areshown in the 𝑥, 𝑦 plane in Fig. 13.7.
The reader is invited to compare the cut lines in figure 13.7 with cutsin maps at the right hand side column of figure suma. The cut lines offunction AuNem𝑞 are the same, as those of function ArqNem𝑞 ; because
it is the first function to evaluate while computing through the asymptotic representation 𝐺𝑞,𝑀 by equation (13.20). The branch points are
determined by function NemBran(𝑞) shown in figure 13.4.
Lines 𝑣 = const in maps of figure 13.7 approach the cut lines in pretty
specific way, that corresponds to decrease of 𝑢, which represents thereal part of the function. This means, that, if one goes along any line𝑣 = const, increasing 𝑢, one approaches to the origin of coordinates,
where the asymptotic representation is accurate, without to cross the
cut lines. This happens due to the specific choice of the inverse function
ArqNem𝑞 , with this goal, the cut lines of function ArqNem𝑞 are chosen
in the function ArqNem. (The default choice of the inverse function,
provided by the routine “Solve” in Mathematica, does not provide this.)
In such a way, the Abel function AuNem for the Nemtsov function isconstructed and implemented. With functions SuNem𝑞 and AuNem𝑞 ,the iterates of function Nem𝑞 can be calculated. They are shown infigures 13.8, 13.9 and described in the next section.

1

2

𝑦=

2

3

𝑛 = −11

𝑛 = −2𝑛 = −3𝑛 = −4𝑥𝑦 = ArqNem2

1

0

𝑛=

2 𝑥)

.1
−0
.2
−0
=
𝑛
3
−0.
𝑛 = .40𝑛 = − .5
0
−
𝑛=

𝑛=

𝑥

𝑥
)
Nem 0(𝑥
q
r
A
=
𝑦

1

3

𝑦 = Nem (m0 (𝑥)

𝑦=

2

.1
−0
𝑛 = −0.2
𝑛 = 0.3−
𝑛 =−0.4
𝑛 = 0.5𝑛=−𝑦 = Ne3𝑛 = 0.
𝑛=0 5
.
𝑛= 4
0 .3
𝑛=
0 .2
𝑛=
0 .1

𝑛=1

𝑦

𝑛=4
𝑛=3
𝑛=2

https://mizugadro.mydns.jp/t/index.php/File:Itnem20plot.jpg

0𝑛=𝑛 = 0.5
𝑛 = 0.4
𝑛 = 0 .3
𝑛 = 0 .2
𝑛 = 0 .1

𝑛=4
𝑛=3
𝑛=2

𝑦

𝑛=1

https://mizugadro.mydns.jp/t/index.php/File:Itnem00plot.jpg

21(𝑥) 𝑛 = −3𝑛 = −2𝑛 = −3𝑛 = −4𝑛Fig. 13.8: 𝑦 = Nem𝑞 (𝑥) at 𝑞 = 0, left, and 𝑞 = 2, right, for various 𝑛 by (13.24)

171

𝑥

4

Iterates of the Nemtsov function

With functions Nem𝑞 and SuNem𝑞 from the previous sections, the iterates of the Nemtsov function can be defined,

Nem𝑞𝑛 (𝑧) = SuNem𝑞(︁)︁
𝑛 + AuNem𝑞 (𝑧)(13.24)

Figure 13.8 shows the explicit plot 𝑦 = Nem𝑞 (𝑥) for 𝑞 → 0 in the left
hand side picture and for 𝑞 = 2 in the right hand side picture, for various
values of the number 𝑛 of iterate. The integer values of 𝑛 correspond to
the thick lines.
For 𝑞 = 1, complex maps of iterates Nem1𝑛 are shown in figure 13.9. The
figure shows, how, at the increase of number 𝑛 of iterate from −1 to
1, function ArqNem1 gradually changes to the identity function (withrectangular grid as the complex map) and then to the Nemtsov function
Nem1 .
Iterates in figure 13.8 look similar to iterates of other quickly growing
holomorphic functions [54, 64, 61, 66, 89]. In particular, at 𝑛 ≈ 0, theiterate Nem𝑞𝑛 looks similar to identical function; at 𝑛 = 1, it is justNemtsov function Nem𝑞 , and at 𝑛 = −1, it is the inverse function, id
est, ArqNem𝑞 .
Iterates of a growing real-holomorphic function are also real-holomorphic;
the complex maps are symmetric with respect to reflection from the realaxis, so, the only upper half of the complex plane is shown shown infigure 13.9. The left column shows maps for the positive iterates; thenumber 𝑛 varies from 0.6 at the top map with step −0.1 to −0.1 at thebottom map. In the similar way, the right hand side column represents
maps for 𝑛 from −0.6 at the top to −0.1 at the bottom. Only case with𝑞 = 1 is presented, but one can download the generator of the figure andplot similar maps for other values of 𝑞 , and, of course, other values ofnumber 𝑛 of iterate; this number can be even complex.
Iterates by 13.24 shown in figures 13.8, 13.9 provide the smooth (holomorphis) transition from the Nemtsov function Nem𝑞 to the identity
function and then to the inverse function ArqNem. Iterates have thegroup property,

𝑇 𝑚+𝑛 (𝑧) = 𝑇 𝑚 (𝑇 𝑛 (𝑧))(13.25)This ratio holds only for certain range of values of parameters, that
includes the positive part of the real axis for 𝑧 . To keep the Book ofreasonable thickness, I skip this analysis and invite Readers to do thisas an exercise.
172

−0.5

𝑦

0.5

0

1

1

0.5
𝑦0

𝑣 = 0 𝑥 0−1 cut

0.5

1 .1
1

0

𝑢=
𝑢
0.5
= 0
𝑢 = −0.
𝑦0
𝑢 = −0.2 1
𝑢 = −0.3
−0
.4
0 cut

cut

𝑣 =1

𝑢=1

𝑦0

0 𝑥0

𝑥 −1
𝑦

0.5

−

1

−0.5

0
𝑢=

𝑦0

𝑦0
cut

0

𝑢=1

𝑣=

0.5

0 𝑥0

0.5

−1
𝑢=

𝑣 =0

𝑣 =1

0 cut
𝑥 −1
−0.5
𝑦
1
.
=1

𝑣

0.5

𝑦0

𝑦0

0

−0.5

cut

0 cut
−1

𝑢=1

0.5

𝑣=

0 𝑥0

0.5

𝑣 =0

0 cut
𝑥 −1

𝑛
Fig. 13.9: 𝑢+i𝑣 = Nem1 (𝑥+i𝑦) for 𝑛 = 0.6 ..
right, by (13.24)

173

𝑣=

0.5

𝑣

0 𝑥0

𝑣 =1

−0.5

0 𝑥0

𝑥

𝑥

0 .6
𝑣 = = 0 .5
𝑣
.4
𝑣 =0
𝑣 = 0.3
𝑣 = 0.2
𝑣 = 0.1
𝑣 =0

𝑢 = 0 .2
𝑢 = 0.1
𝑢 = 0 0.1
𝑢 = − 0.2
𝑢 = − 0.3−
𝑢 = −0.4
𝑢 = −0.5
𝑢 = −0.6
𝑢 = −0.7
𝑢 = −0.8
𝑢=

𝑢=0

−0.5

0.5

0 𝑥0

1.1
𝑣 = =1
𝑣

𝑣 =1

0.5

0 cut
−1
𝑦

−0.5

0.2
𝑢 = 0.1
𝑢 = 0 .1
𝑢 = −0 .2
𝑢 = −0 .3
𝑢 = −0 0.4
𝑢 = − 0.5
𝑢= =−
𝑢

𝑢=

0 cut
−1
𝑦

0 𝑥0

cut

𝑢=

𝑣=

𝑣=

𝑣=

0

1

0.5

−0.5

𝑦

1

0 𝑥0

0.2 .1
𝑢= =0
𝑢

𝑢=

−0.5

𝑣 =0

0 .6
𝑣 = = 0 .5
𝑣
.4
𝑣 =0
3
𝑣 = 0.
𝑣 = 0 .2
𝑣 = 0.1
𝑣 =0

𝑢 = 0.30
𝑢 = .20
𝑢 = .10𝑢=−0.1
𝑢 = −0.2

𝑣=

1

0 cut
−1
𝑦

𝑥

0 .5
𝑣 = 0 .4
𝑣=
0.3
𝑣=
2
𝑣 = 0.
𝑣 = 0.1
0.5

cut

cut

𝑢=

𝑢=1

𝑦0

𝑣 =0

6
0.

0 𝑥0

𝑢=

𝑣=

𝑢=

𝑣=

cut

𝑣 = 0 𝑥 0−1 cut

0.5

𝑣=

0 .3

𝑢=0
1
𝑢 = −0.

1.1

0 𝑥0

𝑣 =0

𝑣 = −1

−0.5

0 .4

.2
𝑢 = 0.1

𝑦0

0 cut
−1
𝑦

𝑢=

0.4
𝑣=
0.3
𝑣=
2
𝑣 = 0.
𝑣 = 0.1

0.5

𝑢=0

0.5

𝑣=

1

1

𝑢=

0.5

0 𝑥0

cut

𝑣=

𝑢=1

𝑦0

0

cut

𝑢=

𝑢=

0.5

−0.5

cut

1

𝑣 =0
0 cut
𝑥 −1
𝑣= 𝑦
2

0.5

𝑣=

cut

𝑢=1

0 𝑥0

𝑣 =1

−

𝑢 = 0.21
𝑢 = 0.
0
𝑢=

𝑦0

.1

−0.5

.4

𝑢 = 0.3

0.5

1

𝑣 =1

0 cut
−1
𝑦

𝑣=

𝑣=

cut

1

𝑢=0

𝑢=0

𝑣 =1

0

𝑣 =0

𝑣=

𝑣=

2

𝑦0

𝑣=

0.5

1

𝑢 = −1

𝑢 = −2𝑣 =−0 .6

https://mizugadro.mydns.jp/t/index.php/File:Nem1it.jpg
𝑦
𝑦

𝑥

8
= 0.

6
𝑣 = 0 . .5=𝑣 0𝑣 = 0.4
𝑣 = 0.3
𝑣 = 0.2
𝑣 = 0.1
𝑣 =0
0.5

𝑥

𝑣 = 0.8𝑣 = 0.6𝑣 = 0.5𝑣 = 0.4
𝑣 = 0.3
𝑣 = 0.2
𝑣 = 0.1
𝑣 =0
0.5

𝑥

0.1, left, and for 𝑛 = −0.6 .. −0.1,

End of exotic iteration

The special 4th order polynomial (13.1) is considered in thissection. I call it Nemtsov function after Boris Nemtsov, Fig.
13.10; the need to denote this function coincided with thetragic event. This polynomial is treated as transfer function:
the inverse function ArqNem, the superfunction SuNem andthe abelfunction AuNem are constructed.

https://nemtsov.ru

5Fig. 13.10:
B.Nemtsov

Construction of function ArqNem happened to be non-trivial; so, its map
is shown in the right hand side column of figure 13.2 for various valuesof parameter. Choice of the cut lines of this function is important. The
readers are invited to try to construct abelfunction with other choice ofthe cut lines and plot the complex maps of the result. And compare theresulting complex maps to those in figure 13.7. And the same for theiterates in figure 13.9.
Function Nem is my last attempt to construct a difficult-to-iterate growing real-holomomorphic function with real fixed point. The fixed point ischosen at zero, because the update to the more general case is straightforward, it is specified in the last row of table 3.1.
For the real-holomorphic growing transfer function with real fixed point,
the suprfunction can be constructed with regular iteration considered inchapter 6, if the derivative at fixed point is not unity, or with exotic
iteration, if this derivative is unity. Then, one can guess the heuristic“solution” with correct asymptotic behaviour of the superfunction, using
analogy with the differential equation, discussed in the previous chapter;the same analogy works for the Nemtsov function too. This leads toiterates that I call “exotic”. The exotic iterates lead to pretty regularand real-holomorphic superfunctions, abelfunctions and correspondingly
regular non-integer iterates of the transfer function.
In the next chapter, I consider, in some sense, even more exotic case,when the transfer function has no real fixed point. The example ofsuch a function without real fixed point is just natural exponent, 𝑇 =exp. Historically, namely this transfer function had been considered anditerated (it can be iterated any real or even complex number of times)
[54].

174

Chapter 14
Natural tetration tet

𝑦 = tet(𝑥)𝑓 (𝑧 +1) = exp(𝑓 (𝑧)) (14.1)

𝑦In order to narrow the set ofsolutions, the additional condition is assumed:

4(14.2)

3e𝑓 (0) = 1

𝑦 = e𝑥https://mizugadro.mydns.jp/t/index.php/File:TetPlotU.png

Here, I consider the exponential transfer function, 𝑇 = exp.
For this transfer function, thetransfer equation can be written as follows:

In order to provide the unique2
ness, in addition, I require,that the solution 𝑓 (𝑧) is holomorphic at the whole complex
1(︀)︀
plane except 𝑧 ≤ −2, and also
𝑦 = 10 tet(𝑥) − (𝑥+1)𝑦 = e𝑥limited at least in the strip
0ℜ(𝑧) ≤ 1. I refer the solu𝑥−1
0
1tion 𝑓 as “tetration” (or “nat(︀)︀
𝑦 = 10 tet(𝑥) − (𝑥+1)ural tetration”) and denote it−1with symbol tet. In this chap𝑦 = tet(𝑥)ter I tell, how this function isconstructed; I use the main formulas and pictures from publi- Fig. 14.1:(︀ 𝑦 = tet(𝑥) , 𝑦)︀= exp(𝑥) , and𝑦 = 10 tet(𝑥) − (𝑥+1)
cations [54, 64].
175

Colleagues ask questions not only about superfunctions and way of theevaluation, but also about guessing of properties of these functions. In
particular, about the natural tetration: “How did you guess?”
The questions about guessing are important not only for the history, butalso for the colleagues, who want to use the similar way for other transfer
functions, superfunctions and, perhaps, even to some more complicatedobjects. So, I consider these questions seriously. Especially, this apply tothe natural exponent and the natural tetration, as the natural tetration
is first non-trivial superfunction, for which the efficient algorithms ofevaluation had been suggested and described [54, 64].
In this chapter, I provide not only the formulas and pictures for thenatural exponental and the natural tetration, but also explanations, why
namely this tetration should be considered and recognised as the most“true” and the “most natural” among various possible superfunctions;and why any researcher, following the same idea, should come to thesame tetration.

1

Exponent

Before to deal with solution 𝑓 of equations (14.1), (14.2), it worth toremind properties of exponent. For the real argument, graphic of exponent is shown in figure 14.1 with thin line. Complex map of the transfer
function 𝑇 = exp is shown in figure 14.2.
I hope to be not condemned for drawing so elementary functions as
exponent. I did not do it in the previous section about sin; but for someelementary functions I provide the maps. The Book should allow theunderstanding, even if it happens to be in hand of a pure experimentalist.
With the detailed descriptions of the elementary things, I hope, even
Akira Shirakawa and Yulya Kuznetsova can understand at least themain idea of the Book. The Book must allow the reading even by theleast-educated academician [16] 1 .
The exponent can be considered as superfunction of function “multipli1 https://royallib.com/read/Strugatsky_Arkady/Tale_of_the_Troika.html Tale of the

Troika by Arkady and Boris Strugatsky. PROLOG. "..Our slogan is ‘elevators for everyone.’
No matter who. The elevator must be able to withstand the entrance of the least-educated academician.”

176

2

𝑣 =0

𝑣 =1
𝑣 =2

3

𝑢 = −1

𝑦

𝑢=0
𝑣 =0

𝑢 = −1 𝑣 = −1 𝑢 = 1
𝑢=20

𝑢 = 0.2

1

−1
−2

𝑢=0

−3

𝑣 =0

−4
−4

−3

−2

−1

Fig. 14.2:

0

1

2

3

𝑥

𝑢+i𝑣 = exp(𝑥+i𝑦)

cation by constant number e”; e = exp(1) ≈ 2.71828182846 . In such away, exp is superfunction for the transfer function 𝑇 by

𝑇 (𝑧) = e 𝑧(14.3)

I repeat the formula from the school course of algebra:

𝑓 (𝑧 +1) = e 𝑓 (𝑧)(14.4)The solution of this equation can be constructed with regular iteration,in vicinity of the fixed point 𝐿 = 0 of the transfer function 𝑇 by (14.3).
The Reader is invited to make this exercise and check, that the primary
expansion stops at the first term, giving the exact solution 𝑓 = exp at
the first iteration.
177

6
5
4

𝑦=
𝑦 = exp√
(
2 𝑥)
𝑥

7

𝑦 = exp( 𝑥 )

𝑦

3
2
1
−5

−4

−3

−2 −1
1
2
3
√
Fig. 14.3: 𝑦 = exp(𝑥) and 𝑦 = exp
2 (𝑥)

4

5

𝑥

For the transfer function 𝑧 ↦→ e 𝑧 , I write also the Abel equation:𝑔(e 𝑧) = 1 + 𝑔(𝑧)(14.5)

Equation (14.5) is considered by Henryk Trappmann [87]. The Reader
can guess, that Henryk got the natural logarithm as the solution, id
est, 𝑔 = ln. Readers are invited to think, what additional requirements
should be associated with equation (14.5), in addition to equation𝑔(1) = 0(14.6)in order to provide the uniqueness of the solution 𝑔 = ln.
Explicit plot of exp is shown in figure 14.3. (The same dependence isshown with thin curve in figure 14.1.) It is worth comparing the graphic√of the natural exponent to that of the exponent to base 𝑏 = 2, whichcrosses the straight line 𝑦 = 𝑥. This line is also shown in figure 14.3.
The graphic of the natural exponent 𝑦 = exp(𝑥) does not cross the line𝑦 = 𝑥. The natural exponent has no real fixed point.
Fixed point 𝐿 of exponential and logarithm to base 𝑏 = exp(𝑎) is solutionof equation 𝐿 = log𝑏 (𝐿). This solution can be expressed through theTania function, considered above in chapter 4:178

𝐿 = filog(𝑎) =Tania(ln(𝑎)−1−𝜋i) WrightOmega(ln(𝑎)−𝜋i)=(14.7)−𝑎−𝑎

The second equality in formula (14.7) can be considered as definition of
the new function filog. This function is described also in TORI,
https://mizugadro.mydns.jp/t/index.php/Filog.
Function filog is considered in details below, in chapter 18. Here, weneed this function only for the single value of the argument, namely,
𝐿 = filog(1).
As function Tania is already described, it should be considered as special
function. In this sense, quantity 𝐿 should is exact:

𝐿 = −Tania(−1−𝜋i)≈ 0.3181315052047641353 + 1.3372357014306894089 i (14.8)
The rough approximation (with two significant figures) for 𝐿 by (14.8)
can be found even from figure 5.2, counting isolines with finger.
Function Tania is used in this Book already twice: first, in chapter 2,
as realistic superfunction for the transfer function Doya (that describes
increase of the intensity of light in a laser amplifier with simple modelof continuously pumped of active medium) and, second time, here, as
exact representation for the fixed point of exponent. This is the samefunction. Recycling, reusing of the results is base of any science, andthe physical mathematics (see the Introduction) is not an exception.
Exponent is real-holomorphic function, exp(𝑧 * ) = exp(𝑧)* ; hence 𝐿* ≈0.1−1.3 i is also the fixed point. In principle, each of these fixed points
can be used for the regular iteration, considered in chapter 6. However,such iteration does not lead to the real-holomorphic superfunction. I
wanted to suggest a way of evaluation of the real-holomorphic tetration,that could be used as the definition. In order to specify it, I assume, that
there exist some special superfunction of the natural exponent, and thissuperfunction is characterised with specific behaviour. As the real partof the argument goes to −∞, the superfunction approaches 𝐿 in theupper part of the complex half plane, and 𝐿* in the loser part. At the
beginning, this is nothing more, but just guess. In the next section, thisguess is used to construct both, the definition and the way of evaluation.
179

2

Heuristic tetration

Since publication of the first article about real-holomorphic tetration,
the colleague ask me similar questions:“Why did you interested in holomorphic tetration?"“How did you guess the asymptotic behaviour to tetration at ±i∞?”“How did you guess the initial approximation fit3 ?” [88].
In order not to be like Rip van Winkle 2 , revealing new and new details,
here I mention the only one of motivations, that is related to physics.
Initially, I wanted to use some fast-growing function in order to represent
the factorial of the number of particles in the Bose-Einstein condensate,
this factorial appears at the attempt to write-out the first approximation
for the normalized multi-particle wave function. The fast-growing function is described in the article by Hooshmand [49], but it happened to be
not suitable for the asymptotic analysis, because it is not holomorphic.
The corresponding extension

⎧ (︀
)︀
⎪
ln
uxp(𝑧
+1)
⎨
𝑓 (𝑧) = uxp(𝑧) =
𝑧+1
⎪
(︀
)︀
⎩
exp uxp(𝑧 −1)

atℜ(𝑧) ≤ −1
at −1 < ℜ(𝑧) ≤ 0
at 0 < ℜ(𝑧)(14.9)

has many cut lines; they divide the complex plane to almost separated
strips. Complex map of function uxp by (14.9) is shown in the upper
picture of figure 14.4 with lines of constant log amplitude 𝑢 and lines of
constant phase 𝑣 ,exp(𝑢+i𝑣) = 𝑓 (𝑥+i𝑦)(14.10)This representation is different from that, used for the most of complex
maps in this Book; usually, the lines of constant real part and thoseof constant imaginary part are drawn. While I explain, how did I get
the holomorphic tetration, I represent maps in the same form, as they
appear in the original paper [54].
As I already mention above, the vertical cuts of the range of holomorphism of function uxp divide the complex plane to almost independent
strips seen in the top map in figure 14.4. These strips raise the question:
Is it possible to suggest a “more holomorphic” (id est, with less cuts) solution of the transfer equation (14.1)? Or no solution 𝑓 of the equation(14.1) may have wide range of holomorphism?
2 https://classiclit.about.com/library/bl-etexts/wirving/bl-wirving-rip.htm

Van Winkle by Washington Irving. (1783-1859)
180

Rip

−1

𝑢

−8

𝑦
1
0

−6

−4

−2
𝑣 = 1.4

𝑢 = 1.2b 𝑣 = 1.4

𝑢 = 0.4

𝑣 = 1.2
𝑣 =1

𝑦
3
2

−6

−4

−2

0

c

𝑣 = 1.4
𝑢 = 0.4
𝑣 = 1.2

𝑣 = 1.4

2

4

6

8 𝑥

𝑣 = 1.4

𝑢 = 0.4
𝑣 = 1.2𝑣 =1

𝑢=1

𝑢 = 0.4−10
𝑢 =𝑣 = −1

𝑣 = −1.4
𝑢 = 0.4−20
𝑢 =𝑣 = −1

𝑣 = −1.4
𝑢 = 0.4−3

𝑢+i𝑣 ≈ 𝐿 ≈ 0.3 − 1.3i

−6

−4

−2

𝑢=

𝑣 = −1.4*−8

𝑢 = 0.4

0

2

4

6

𝑣 = 1.4

𝑢 = 0.4
𝑣 = 1.2
𝑣 =1

0

8 𝑥

𝑣 = 1.4d𝑢+i𝑣 ≈ 𝐿 ≈ 0.3 + 1.3i𝑣 = 1.41

𝑢 = 0.4
𝑣 = 1.2

𝑣 = 1.4

𝑢 = 0.40

𝑢 = 0.4
𝑣 = 1.2𝑣 =1𝑣 =0

𝑢 = 0.4−10
𝑢 =𝑣 = −1

𝑣 = −1.4
𝑢 = 0.4

−2
−3

8 𝑥

𝑢 = 0.4
𝑣 = 1.2
𝑣 =1

0

2

6

𝑣 = 1.2𝑣 =1

𝑢+i𝑣 ≈ 𝐿 ≈ 0.3 + 1.3i

𝑢 = 0.43

4

𝑣 = 1.41𝑦20
𝑢 =𝑣 = −1

𝑢 = 0.4

−8

0

𝑣 =0

𝑣 = −1.4

−1

𝑣 =0
=0

𝑢=1

𝑣 =1

𝑣 =1

𝑢=1

0

a

1

1

𝑣=

𝑦

0
𝑢 =𝑣 = −1

𝑣 = −1.4
𝑢 = 0.4

𝑢+i𝑣 ≈ 𝐿 ≈ 0.3 − 1.3i

−8

−6

−4

−2

𝑢=

𝑣 = −1.4*

𝑢 = 0.4

0

2

4

6

https://mizugadro.mydns.jp/t/index.php/File:Analuxp01u400.jpg
Fig. 14.4:

0

8 𝑥exp(𝑢+i𝑣) = 𝑓 (𝑥+i𝑦) for the following functions: 𝑓 = uxp, (a);𝑓 = Fit3 , (b); 𝑓 = Fit6 , (c); 𝑓 = tet, (d)181

At the first look into results by Hooshmand [49], construction of thereal-holomorphic solution seemed to be impossible. On the other hand,
the initial assumption used there, about monotonous derivative (for thereal argument) of the superfunction of exp, looks doubtful. I tried toconstruct some alternative proof, that the cuts are unavoidable, without
to use the strange assumption. I begun to investigate the case, assuming
existence to the holomorphic solution. I expected to get some contradiction, and to use the contradiction for the proof. The holomorphic
solution had been constructed, and no contradiction had been detected[54]; so, I had to accept the existence.
Historically, the construction of this solution begun with approximations. I had considered several real-holomorphic elementary functions,that have logarithmic singularity at −2 and take the same values, as
tetration, at few integer values of the argument. One of them (thathappened to be better than some others) is

fit2 (𝑧) = ln(2+𝑧)
(︁
)︁
)︁
(︁
)︁(︁
𝑧
4
+(1+𝑧) 1 + 2 exp (𝑧 −1)𝑠2 (𝑧) e− 2+ln 3 − ln 2

(14.11)

where

(︁

)︁
𝑠2 (𝑧) = exp exp(𝑧 − 2.51) − 0.6 + 0.08(𝑧 +1)(14.12)

Constants in the expression (14.12) are chosen in order to minimise theresidual at the substitution 𝑓 = fit2 into the transfer equation (14.1).
This approximation could be improved, comparing (14.12) with theprecise approximation through the Cauchy integral, considered below.
However, at the heuristic search for the rough approximations, the representation through the Cauchy Integral had not yet been written; so,choosing the approximation, I had to use the residual as the criterion.
After construction of function fit2 , it happened, that the linear combination of functions 𝑧 ↦→ fit2 (𝑧) and 𝑧 ↦→ ln(fit2 (𝑧 +1)) gives the residual
even smaller; in such a way, the approximation fit3 appeared:(︀)︀
fit3 (𝑧) = 0.6 fit2 (𝑧) + 0.4 ln fit2 (𝑧 + 1)

(14.13)

The range of approximation of tetration can be extended. Let

⎧ (︀
)︀
⎪
ln
Fit
(𝑧
+1)
3
⎨
Fit3 (𝑧) =fit3 (𝑧)⎪(︀)︀⎩
exp Fit3 (𝑧 −1)

atℜ(𝑧) ≤ −1
at −1 < ℜ(𝑧) ≤ 0
at(14.14)0 < ℜ(𝑧)Logampliture and phase of this function are show in figure 14.4b.
182

For comparison, two more maps are shown in figure 14.4, they are
numbered as c and d. There represent the asymptotic approximation,
𝑓 = Fit6 and tetration 𝑓 = tet, described below in section 4; namely
this tetration is goal of this chapter.
The asymptotic approximation{︃
Fit6 (𝑧) =𝐿 + exp(𝑘(𝑧+𝑟))(︁)︁ , ℜ(𝑧) < −8
exp fit6 (𝑧 −1) , ℜ(𝑧) ≥ −8

(14.15)is good at large values of imaginary part of the argument. For natural
tetration, the increment 𝑘 = 𝐿. This looks as just coincidence. However,everyone can check it with the asymptotic analysis, substituting the
primary expression of fit6 into the transfer equation (14.1). Value of
constant 𝑟 ≈ 1.075820830781−0.9466419207254 i appears as adjusting
parameter of this approximation. On the other hand, it seems to beapproximation of the important mathematical constant. I would call this𝑟 with name “The Kneser constant”. This is one of constants, required
for the expansion of iterates of exponent discussed in [10] and used forthe approximation of tetration [64].
Function Fit6 (𝑧) approximates tetration tet(𝑧) at ℑ(𝑧) > 0.4; function
Fit6 (𝑧 * )* approximates tet(𝑧) at ℑ(𝑧) < −0.4; combination of thesefunctions is shown in map “c” in figure 14.4. In vicinity of the real axis,
roughly, in the strip |ℑ(𝑧)| < 0.4, both these functions Fit6 (𝑧) and
Fit6 (𝑧 * )* look ugly, and this strip in the map is left white.
Approximations 𝑓 = Fit3 and 𝑓 = Fit6 by (14.14) (14.15) are alreadysufficient to plot the complex maps and explicit plots of tertration; together, they provide of order of 3 decimal digits in the range of maps
shown in figure 14.4. The last map "d" in figure 14.4 visually looks as
superposition of the maps "b" and "c" above; this gave the general viewof tetration that had to be constructed.
The approximations above (even Fit3 ) allow to guess the asymptotic
behaviour of tetration. It should approach the fixed points 𝐿 or 𝐿*
of logarithm, while the imaginary part of the argument approaches theplus or minus infinity. These values are indicated in maps “c” and “d” offigure 14.4.
In such a way, this section explains, how did I guess, which the asymptotic behaviour should the tetration have. Postulating this behaviour,
one can construct the algorithm for evaluation of tetration with any required precision. The postulated properties of tetration are collected in
the next section.
183

3

Properties of tetration

Following recommendations by colleagues, friends and relatives, this content of this Book gradually goes from the simple examples to the moregeneral formulas. In order to follow this way, in this chapter, I define
only the natural tetration. This section continues following article [54]of year 2009. .
Since the two upper maps in figure 14.4 were plotted, the main propertiesof this function are clear. I postulate them below.
Solution 𝐹 of the transfer equation (14.1) with additional condition
𝐹 (0) = 0 is called natural tetration, or simply tetration tet, if the following conditions are satisfied:
Function 𝐹 (𝑧) is real-holomorphic in the whole complex plane
except the halfline 𝑧 ≤ −2. Id est, 𝐹 (𝑧 * ) = 𝐹 (𝑥)* . At 𝑧 = −2, function
𝐹 (𝑧) has logarithmic singularity, id est, the branch point.
Т1.

Т2. Function 𝐹 (𝑧) is bounded in the strip |ℜ(𝑧)| ≤ 1.
T3. Function 𝐹 (𝑥) asymptotically approaches the fixed point 𝐿 in theupper half plane: for any real 𝑥, the relation below holds:

lim 𝑓 (𝑧 + i𝑦) = 𝐿

𝑦→+∞(14.16)

In addition, for positive 𝑦 , the relation below holds:

lim 𝑓 (𝑧 + i𝑦) = 𝐿

𝑥→−∞

T4. In the strip(14.17)−1 ≤ ℜ(𝑧) ≤ 2 , the following condition holds:
arg(𝐹 (𝑧)) < 2(14.18)

Conditions T1-T4 above are a little bit redundant. The following development of the formalism of superfunctions is expected to indicate, whichof these properties should be kept as definition of tetration, and whichshould appear as theorems, following from the shortened definition.
From postulates (14.16), (14.17) and real holomorphism 𝑓 (𝑧 * ) = 𝑓 (𝑧)* ,it follows, thatlim 𝑓 (𝑧 + i𝑦) = 𝐿*𝑦→−∞(14.19)and for negative 𝑦 , the relation below holds

lim 𝑓 (𝑧 + i𝑦) = 𝐿*𝑥→−∞(14.20)

These conditions are used in the next section for construction and evaluation of tetration through the Cauchy integral.
184

4Cauchy integral

For holomorphic function 𝐹 , the Cauchy formula takes place [105]:1
𝐹 (𝑧) =
2𝜋i

∮︁

𝐹 (𝑡)
d𝑡
Ω 𝑡−𝑧

(14.21)

where contour Ω belongs to the simply connected range of holomorphism of function 𝐹 and only once passes around point 𝑧 in the counterclock wise direction. With equation (14.21), the transfer equation (14.1)
leads to the integral equation for the values of superfunction along theimaginary axis [54]. The description is repeated below.
Engineer, physicist or mathematician, using formula (14.21), has certain
freedom in choice of the contour of integration. The researcher acts ina way, similar to that of an Engineer, who makes a project of the loop
railroad for the rocky semi-island.
The smart engineer takes into account the locations of ports, cities,
farms, industries, in order to help the people to reach the places of
their destination. Also, the engineer tries to avoid swamps, steep slops,
narrow curvy canyons, to make the railroad fast, cheap and safe.
Vainglorious tyrant, dictator, already famous in sports, war, art, archeology and ornithology, who wants to show himself also as a powerful
all-mighty engineer, may draw a rectangle on the map, and promote it
as a project of the trace of the railway. Such a “ project ” will require a
lot of bridges, ramps, excavations and tunnels, makes him famous also
as vain waster of the state budget and may bring him to the situation“no money” 3 .
Sorry, in the choice of the contour of integration, described in [54], I look
like as a tyrant, rather than as a smart engineer: I choose the contour of
integration in the shape of rectangle. The only excuse is, that the thiscontour leads to efficient way of evaluation of tetration.
Let 𝐹 be real-holomorphic of equation (14.1);
Let 𝐴 be real positive number, so big, that 𝐹 (i𝐴) ≈ 𝐿
Let the range of hlomorphizm of function 𝐹 (𝑧) includes the domain−1 ≤ ℜ(𝑧) ≤ 1 and, in this range, let | arg 𝐹 (𝑧)| < 𝜋 .
3 https://weirdrussia.com/2016/05/28/meme-medvedev-says-we-have-no-money-but-you-hang-in-there/

Medvedev Says “We have no money, but you hang in there” (2016).

https://www.bbc.com/news/blogs-trending-36482124 Russian PM: ’No money for pensions,but have a good day!’ 2016.06.09
https://www.cnbc.com/2016/06/09/there-is-no-money-left-bye-russian-pm-causes-social-media-storm.html

Holly Ellyatt. ’There is no money left, bye!’: Russian PM causes social media storm. .. "nomoney left" in Russia’s budget.. "There just isn’t any money now. .."185

These conditions allow to convert the contour integral into the “solvable”
integral equation; of course, at the appropriate choice of the contour of
integration. Let contour Ω consists of the four segments:
A. Segment along line ℜ(𝑡) = 1 from 𝑡 = 1−i𝐴 to 𝑡 = 1+i𝐴.
B. Segment from point 𝑡 = 1+i𝐴 to 𝑡 = −1+ i𝐴, passing above point 𝑧 .
C. Segment along line ℜ(𝑡) = −1 from 𝑡 = −1+ i𝐴 to 𝑡 = −1−i𝐴.
D. Segment from point 𝑡 = −1−i𝐴 to 𝑡 = 1−i𝐴, passing below point 𝑧 .
For this contour Ω, the Cauchy integral can be written as follows:

1
𝐹 (𝑧) =2𝜋

∫︁ 𝐴

1
𝐹 (1+i𝑝) d𝑝−
2𝜋−𝐴 1 + i𝑝 − 𝑧∫︁ 𝐴

𝐹 (−1+i𝑝) d𝑝−𝐴 −1 + i𝑝 − 𝑧(14.22)

−

𝐹up
2𝜋i

∫︁ 1−i𝐴

d𝑡
𝐹down
+
2𝜋i
−1−i𝐴 𝑡−𝑧∫︁ −1−i𝐴
−1−i𝐴

d𝑡
𝑡−𝑧

where 𝐹up и 𝐹down are some mean values of function 𝐹 in vicinity of thesegments B and D of the contour Ω.
Taking into account the transfer equation 2.12, and assuming holomorphism of function 𝑇 −1 , equation (14.22) can be rewritten as follows:

1
𝐹 (𝑧) =
2𝜋

(︀
)︀
(︀
)︀
∫︁ 𝐴
exp 𝐹 (i𝑝) d𝑝
ln 𝐹 (i𝑝) d𝑝
1−+ 𝐾(𝑧) (14.23)
1 + i𝑝 − 𝑧
2𝜋 −𝐴 −1 + i𝑝 − 𝑧
−𝐴

∫︁ 𝐴

where

(︂

)︂
1
1
1 − i𝐴 + 𝑧
𝐾(𝑧) = 𝐹up ·−ln
2 2𝜋i 1 − i𝐴 − 𝑧
)︂
(︂
1
1
1 − i𝐴 − 𝑧−ln+ 𝐹down ·2 2𝜋i 1 − i𝐴 + 𝑧(14.24)This representation implies that the modulus of phase of function 𝐹
along the imaginary axis remains less than 𝜋 , so, the contour of integration does not cross the cut line of the logarithmic function in (14.24).
Equations (14.23),(14.24) are still exact. However, they become approximations, if we replace 𝐹up → 𝐿 and 𝐹down → 𝐿* . This replacement
leads to the closed representation for 𝐾 . We get the “solvable” integral equation for the approximation 𝐹𝐴 (i𝑦) of superfunction 𝐹 along
the imaginary axis:1
𝐹𝐴 (i𝑦) =
2𝜋

(︀
)︀
(︀
)︀
∫︁ 𝐴
exp 𝐹𝐴 (i𝑝) d𝑝
ln 𝐹𝐴 (i𝑝) d𝑝
1−+ 𝐾𝐴 (i𝑦)1 + i𝑝 − i𝑦
2𝜋 −𝐴 −1 + i𝑝 − i𝑦−𝐴
(14.25)

∫︁ 𝐴

186

−3

−2 −1 0

2

3 𝑥

−3

𝜌 = −2−2 −1 0

1

2

3 𝑥

𝜙=0
−3
.2

−4

𝜌=

−1

−2

1

0

1

𝜙=2

1

𝜙=2

𝜌=

−3

𝜌=

𝜌 = −1

−1

1
𝜙=0

0

𝐴 = 10

𝑦
3
2

0

−2

𝜌 = −2

𝜙=

𝜌=

𝐴=5

𝜙=

−1
.6

𝜙=0

−2

−1

2
1

𝜙=0

0

𝜌 = −1

𝑦
3

𝜙=

0 .2

𝜌=

𝜙=2
𝜙=
1

1

𝐴=3

𝜌 = −1

𝜙=3

2

𝜌=

𝜌=0

𝑦
3

−2
−3

𝜌=

−2 −1 0

1

2

−3

3 𝑥

𝜌=0

https://mizugadro.mydns.jp/t/index.php/File:Analuxp02t900.jpg
Fig. 14.5:

exp(𝜌+i𝜙) = 𝐾𝐴 (𝑥+ i𝑦) by (14.26) for 𝐴 = 3, 5, 10

where

(︂
𝐾𝐴 (𝑧) = 𝐿 ·1
1−i𝐴+𝑧1−ln
2 2𝜋i 1+i𝐴−𝑧

)︂

*

+𝐿 ·

(︂

)︂
1
1−i𝐴−𝑧1−ln
2 2𝜋i 1+i𝐴+𝑧(14.26)Equations (14.25) и (14.26) include no unknown parameters; neither 𝐹up
not 𝐹down appear there. In such a way, equations (14.25) and (14.26)
can be used to calculate 𝐹𝐴 .
Representation of the integral by the uppest and lowest pars of thecontour Ω through function 𝐾𝐴 by (14.26) is not trivial. The first lookat the expression causes the seduction to rewrite each logatightm as
difference of two logarithms, in order to simplify the expression in theargument. However, in other to get a robust representation for tetration,
this is not a good idea. On the representation (14.26), the cuts are
directed away from the imaginary axis; they go horizontally, parallel tothe abscissa axis. This helps to avoid approaching of the argument of the
primary approximation of tetration to the boundaries of holomorphismof function 𝐾𝐴 . Complex map of function 𝐾𝐴 is shown in figure 14.5
for 𝐴 = 3, 𝐴 = 5 and 𝐴 = 10 with lines of constant logamplitude 𝜌 andconstant phase 𝜑 such that exp(𝜌+i𝜙) = 𝐾𝐴 (𝑥+ i𝑦) in the 𝑥, 𝑦 plane.
Solution 𝐹𝐴 of equation (14.25) can be approximated with the iterates,
described below. At 𝐴 ≫ 1, solution 𝐹𝐴 provides the good approximation for the superfunction 𝐹 ≈ 𝐹𝐴 ;𝐹 (𝑧) = lim 𝐹𝐴 (𝑧)𝐴→∞(14.27)

In order to get tetration tet, the additional condition tet(0) = 1. should
be satisfied. So, I definetet(𝑧) = 𝐹 (𝑥0 + 𝑧)
187

(14.28)

where 𝑥0 is solution of equation 𝐹 (𝑥0 ) = 1. Value 𝑥0 depends on theinitial condition at the iterations, and also on the order of update of
the values of the discrete approximation of function 𝐹𝐴 . For the initialapproximation by (14.15), the resulting 𝑥0 happens to be or order of
0.1 ; and it is important, that it remains smaller than unity.
Formula (14.28) can be considered as “constructive definition” of tetration tet, with all reverences about the existence and the uniqueness of
the limit, at 𝐴 → ∞ and the limit of infinite increase of the number ofpoints for the discrete approximation of the integrals. From the point ofview of the “pure” mathematics, such a “definition” deserves critics, but
I expect, with time, the “pure mathematicians” present a more efficientand elegant way of constructive definition of tetration. For a while, tetration may remain as solution 𝑓 of the transfer equation e𝑓 (𝑧 ) = 𝑓 (𝑧 +1)
bounded in the strip |ℜ(𝑧)| < 1 with additional condition 𝑓 (0) = 1.
For the approximation of limit in equation (14.27), some finite valueof 𝐴 should be chosen. Value of increment 𝑘 = 𝐿 ≈ 0.318+1.337 i inapproximation (14.15) indicates, that for the complex double implementation, the reasonable value of constant 𝐴 should be of order of 20. It
happened, that for 𝐴 = 24, the residual is a little bit smaller, and thisvalue is used for the primary numerical implementation 4 .
For the numerical implementation of equation (14.25), the integrals are
replaced to their approximations with the Gauss-Legendre quadratureformula. Then, the resulting equation can be solved with iterations,
updating values of the function one by one.
The attempt of the parallel assignment of the new values (that is easy
to program with the high-level programming languages) leaded to thediverging algorithm. In order to get the convergence, I update first the
odd nodes, and then the even ones. Then, after some teens of iterates,
the procedure provides the accurate solution with 14 significant figures;
this precision is estimated, evaluating the residual in various tests of theinternal self-consistency. 5
Solution of equation (14.25) approximates values of superfunction 𝐹
along the imaginary axis. Then, equation (14.23) extends this approxi4 Using the numerical implementation of the Cauchy integral for the first time, I did not guessthe simple estimate through the increment 𝑘 ; so I had to increase value of 𝐴 until the residual at
the substitution of the primary approximation into the transfer equation (14.1) became of orderof the rounding errors of the complex double arithmetics
5 following Axiom 4 (see Introduction), I made certain efforts trying to refute, negate the concepts of existence and uniqueness of tetration.

188

mation to the strip−1 < ℜ(𝑧) < 1(14.29)Accuracy of this primary approximation is poor in vicinity of the edgesof the strip. Therefore, for the numerical implementation of tetration,
the narrower strip is used,11− ≤ ℜ(𝑧) ≤2
2(14.30)(︀)︀𝐹 (𝑧) = 𝑇 𝑛 𝐹 (𝑧 −𝑛)(14.31)applying formulafor some appropriate integer 𝑛, positive or negative, dependently on thesign of ℜ(𝑧).
At the increasing of value of parameter 𝐴 in equation (14.22), function 𝐹
approaches to the solution of the transfer equation (14.1). This solution
does not yet satisfy the condition 𝐹 (0) = 1, but, the required tetration
appears as the appropriate displacement of the argument with equation(14.28).
In figure 14.5, the strip |𝑥| ≤ 0.5, |𝑦| ≤ 4.5 is shaded; roughly, this isregion, where the function 𝐾𝐴 is used at the evaluation of superfunction
𝐹 at the calculation of map in figure 14.4d. The complex maps verify,that the phase of reconstructed function does not exceed 2; and therefore, does not exceed 𝜋 . This justifies use of logarithm in formula (14.23).
However, this cannot be considered as a rigorous proof of existence andthe uniqueness of the resulting function. I hope, the rigorous proof will
be reported soon by the “pure” mathematicians. While I present only
the computational evidence of the existence and the uniqueness.
In general case, applying this method to general transfer function 𝑇 withcomplex fixed points, 𝐹 (𝑧 −𝑛) could happen at the point of singularity
of at the cut line of function 𝑇 or 𝑇 −1 . This would indicate, that theresulting superfunction is also singular. (For tetration, this happens at
the real argument, equal to or smaller than −2.) This may limit therange of applicability of the method suggested here.
In the first calculus, the approximation fit3 had been used as the initialprobe function for the iterational solution of equation (14.25). Thenit happened, that the iterates with other (more primitive) probe functions lead to the same result, providing the same tetration. With thealgorithm above, in 2008, the natural tetration had been constricted189

[54]. For 𝐴 = 24, the Gauss-Legendre quadrature formula with 2048
nodes gives the accurate approximation: at the substitution into equation (2.12), it gives the residual of order of 10−14 , while the variables
complex(double) are used. Of order of 14 significant figures of the solution can be evaluated in real time. This indicates stability of evaluationof tetration through the Cauchy integral.
The algorithm above had been used to plot first maps and explicit plotsof tetration to base 𝑏 > exp(1/e) [106, 107]. In particular, this algorithm is used to plot the map at the bottom of figure 14.4. The direct
representation through the Cauchy integral is fast enough to plot themaps and the explicit plots of tetration in real time. However, it becomes slow, if the tetration is used for evaluation of other function, forexample, its superfunction (pentation) considered below in chapter 19.
For the application of tetration, even faster approximations may havesense. One of them is considered in the next section.

5

Taylor expansion at zeroI wanted the check the claim, that solution 𝑓 = tet of equations (14.1)
and (14.1), that satisfies properties T1-T4, exists and is unique. As it
is declared above, the numerical test does not substitute the rigorous
proof, in the similar way, as the rigorous proof does not substitute thenumerical tests. For the serious tests, it is important, that the function
is fast to evaluate. In order to boost the evaluation, the approximations
through the elementary functions had been suggested [64]. One of themrefers to the Taylor expansion at zero. It is considered in this section.
Derivatives of tetration can be calculated by differentiation of the primary representation by (14.25). More accurate values can be obtainedwith the Cauchy integral formula with the circular contour of integration(assuming, that the primary representation is already implemented).
Radius of this circle should be less than 2, and can be slightly larger
than unity; then, the error of the result reduces due to denominator in
the fraction under the integral in the right hand sidle of equation (14.21).
In such a way, the coefficients in the expansion below are evaluated.

naiv(𝑧) =𝑁−1∑︁

𝑐𝑛 𝑧 𝑛(14.32)𝑛=0tet(𝑧) = naiv(𝑧) + 𝒪(𝑧 𝑁 )190(14.33)

Table 14.1: Coefficients in expansions (14.32), (14.36) and (14.40)

𝑛
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

𝑐𝑛
1.00000000000000
1.09176735125832
0.27148321290170
0.21245324817626
0.06954037613999
0.04429195209047
0.01473674209639
0.00866878181723
0.00279647939839
0.00161063129058
0.00048992723148
0.00028818107115
0.00008009461254
0.00005029114179
0.00001218379034
0.00000866553367
0.00000168778232
0.00000149325325
0.00000019876076
0.00000026086736
0.00000001470995
0.00000004683450−0.00000000154924
0.00000000874151−0.00000000112579
0.00000000170796

𝑠𝑛ℜ(𝑡𝑛 )ℑ(𝑡𝑛 )

0.30685281944005
0.59176735125832
0.39648321290170
0.17078658150959
0.08516537613999
0.03804195209047
0.01734090876306
0.00755271038865
0.00328476064839
0.00139361740170
0.00058758348148
0.00024379186661
0.00010043966462
0.00004090111776
0.00001654344436
0.00000663102846
0.00000264145664
0.00000104446533
0.00000041068839
0.00000016048059
0.00000006239367
0.00000002412797
0.00000000928797
0.00000000355850
0.00000000135774
0.00000000051587

0.37090658903229
0.01830048268799−0.04222107960160−0.01585164381085
0.00264738081895
0.00182759574799
0.00036562994770
0.00002689538943−0.00003139436775−0.00001376358453−0.00000180290980
0.00000026398870
0.00000024961828
0.00000007899707
0.00000000637479−0.00000000341142−0.00000000162203−0.00000000038743−0.00000000001201
0.00000000002570
0.00000000000935
0.00000000000170−0.00000000000005−0.00000000000016−0.00000000000005−0.00000000000001

1.33682167078891
0.06961107694975
0.02429633404907−0.01478953595879−0.00657558130520−0.00025319516391
0.00028246515810
0.00014180498091
0.00003583704949−0.00000183512708−0.00000314787679−0.00000092613311−0.00000013664223
0.00000003171468
0.00000002270476
0.00000000512289
0.00000000031619−0.00000000027282−0.00000000013440−0.00000000002543
0.00000000000045
0.00000000000186
0.00000000000071
0.00000000000012−0.00000000000001−0.00000000000001

Evaluations of coefficients 𝑐 are shown in the first column of table 14.1.
The Taylor expansion tet(𝑧) at 𝑧 = 0 converges for |𝑧| < 2. The radius ofconvergence is determined by the distance from the point of expansion(id est, from zero) to the nearest singularity, which is −2.
For the numerical implementation, the number of terms chosen 𝑁 = 50.
Complex map of the resulting naive approximation by (14.32) is shownin the left hand side of figure 14.6 with lines of constant real part andcontent imaginary part, 𝑢+i𝑣 = naiv(𝑥+i𝑦). The thick lines show levels
𝑢 = ℜ(𝐿) and 𝑣 = ±ℑ(𝐿).
In order to verify the precision of the approximation 𝑓 = naiv, thecentral and the right hand side pictures of figure 14.6 show the maps of

191

𝑦

2

2
0

𝑣=

−1
−2

−2 −1 01

𝐷1 < 11ℑ(
𝐿*)

𝑢 = ℜ(𝐿)

0

𝐿)
ℑ(

1

𝑣=

𝑦

𝑦
1

15

−1

−2

−2
−2 −1 0

15

0

−1

2 𝑥

𝐷2 < 12

1

2 𝑥−2 −1 01

2 𝑥

https://mizugadro.mydns.jp/t/index.php/File:Vladi04.jpg
Fig. 14.6: 𝑢+i𝑣 = naiv(𝑥+i𝑦) by (14.32) for 𝑁 = 50, left; agreements 𝐷1 = 𝐷naiv1 (𝑥+i𝑦)
and 𝐷2 = 𝐷naiv2 (𝑥+ i𝑦) by (14.34) and (14.35), central and right hand side maps.

agreements

(︂

)︂

(︂

)︂

| ln(naiv(𝑧 +1) − naiv(𝑧)|𝐷naiv1 (𝑧) = − lg
| ln(naiv(𝑧 +1)| + |naiv(𝑧)|| exp(naiv(𝑧 −1) − naiv(𝑧)|𝐷naiv2 (𝑧) = − lg
| exp(naiv(𝑧 −1)| + |naiv(𝑧)|(14.34)(14.35)Functions of agreement 𝐷 indicate, how many significant figures of tetration can be expected to appear at evaluation of tetration with approximation “naive” by (14.32). Levels 𝐷 = 1, 2, 4, 6, 8, 10, 12, 14 are shown.
Level 𝐷 = 1 is shown with thick lines. Symbol “15” indicates the domain,
where the agreement is better than 14. We may expect, at |𝑧| < 1, thepolynomial by (14.32) provides of order of 14 significant figures; this is
close to the maximal precision for variables complex double.
Evaluations with 50 terms is considered for verification of the expansion.
At the evaluation of tetration, for example, at the implementation forreal argument, the number of terms can be significantly reduced without
loss of precision.
The polynomial approximation naiv by (14.32) can be used for the precise and fast evaluation of tetration, while the modulus of its argument
is smaller or of order of unity. For the efficient implementation, this isgood, but it is not sufficient. In the next section, the advanced expansion is considered, that allows to extend the range of the approximationof tetration for moderate values of the argument.

192

𝑣=

𝑦
2

𝑢=1

𝑢 = ℜ(𝐿)

−1

ℑ(𝐿 *
)

−1

1

)

0

ℑ(𝐿

𝑣=

1

𝑣=

𝑦
2
1
0

𝑦
2
1

𝐷3 > 140−1

𝐷4 > 14−1−2
𝐷3 < 1−2 −1 0 1 2 𝑥−2 −1 0 1 2 𝑥−2 −1 0https://mizugadro.mydns.jp/t/index.php/File:Vladi05.jpg
𝑣=−2Fig. 14.7:−2

𝐷4 < 1
1 2 𝑥

u+i𝑣 = maclo(𝑥+i𝑦) by (14.36) at 𝑁 = 101, left; agreements

𝐷3 и 𝐷4 by (14.38) и (14.39), centre and right6

Improved approximation

The range of the accurate approximation of tetration can be extended, if
we take into account the logarithmic singularity of tetration. I “switchout” the singularity at −2, expanding function tet(𝑧) − log(𝑧 + 2) instead of tet(𝑧). This expansion gives the approximation below, I call it“maclo”:𝑁−1∑︁𝑠𝑛 𝑧 𝑛 ;(14.36)tet(𝑧) = maclo(𝑧) + 𝒪(𝑧 𝑁 ) .

(14.37)

maclo(𝑧) = ln(𝑧 +2) +𝑛=0

For 𝑛 = 101, function maclo is shown in the left map of figure 14.7.
The series, used for approximation (14.36) converges at |𝑧| < 3; thefunction reproduces the logarithmic branch point and even part of the
cut at 𝑧 < −2. Approximate values of first coefficients 𝑠 are shown inthe second column of table 14.1.
The range of approximation of tetration tet with function maclo is wider,than that by the Taylor expansion of tetration at zero; compare figure
14.6 and figure 14.7. The central and right hand side maps of figure
show agreements⃒ ⎞⎛ ⃒⃒ (︀)︀⃒⃒ ln maclo(𝑧 +1) − maclo(𝑧)⃒⃒⎠𝐷3 (𝑧) = − lg⎝ ⃒⃒ (︀)︀⃒⃒ ⃒⃒⃒⃒ ln maclo(𝑧 +1) ⃒ + ⃒maclo(𝑧)⃒
⃒ ⎞
⎛ ⃒⃒
(︀
)︀
⃒
⃒ exp maclo(𝑧 −1) − maclo(𝑧)⃒⃒⎠𝐷4 (𝑧) = − lg⎝ ⃒⃒(︀)︀⃒⃒ ⃒⃒⃒⃒ exp maclo(𝑧 −1) ⃒ + ⃒maclo(𝑧)⃒193(14.38)(14.39)

Within the central loop, the residuals at the substitution 𝑓 → maclo into
equations (14.1) are of order of 10−15 .
While |𝑧| < 2, the approximation maclo(𝑧) with a hundred terms provides of order to 14 significant figures of tetration tet(𝑧). However, while
the module of the argument increases and becomes larger than two, theaccuracy of this approximation quickly drops down. In order to even extend the range of the fast approximation, the Taylor expansion at somepoint at the imaginary axis can be used. In the next section, the Taylor
expansion at point 3 i is described.

7

Expansion of tet(𝑧) at 𝑧 = 3 i

For evaluation of tetration, we should cover some strip of unity widthalong the imaginary axis with good (fast and precise) approximations.
The approximation maclo from the previous section does not approximate tetration at point 3 i. For me, this is sufficient reason (or, may be,a pretext) to prepare the Taylor expansion of tetration namely in thispoint 6 . This expansion is described below.
The truncated Taylor expansion of tet(𝑧) at point 𝑧 = 3 i is denoted withname “tai” (TAylor expansion centered at the Imaginary axis):

tai(𝑧) =𝑁−1∑︁

𝑡𝑛 (𝑧 −3 i)𝑛(14.40)𝑛=0Approximations of the coefficients 𝑡 are calculated with the Cauchy integral. The real and imaginary parts of the first coefficients are presented in the last two columns of table 14.1. The series converges at√√|𝑧 − 3i| < 22 +32 = 13 ≈ 3.6 . For the numerical implementation I
choose value 𝑁 = 51; then, at |𝑧−3 i| < 2, approximation tai by (14.40)
provides of order of 14 significant figures. The complex map or thisapproximation is shown in the left hand side of figure 14.8.
The right hand side map in figure 14.8 shows the agreement⃒ ⎞⎛ ⃒⃒ (︀)︀⃒⃒ ln tai(𝑧 +1) − tai(𝑧)⃒⃒⎠𝐷5 (𝑧) = − lg⎝ ⃒⃒ (︀)︀⃒⃒ ⃒⃒⃒⃒ ln tai(𝑧 +1) ⃒ + ⃒tai(𝑧)⃒(14.41)

6 Constant 3 appears as minimal integer number for which (with coefficient i) approximation

maclo by (14.36) fails.
194

𝑦
6

𝑦
6
𝐿)
𝑣 = ℑ(

2

(𝐿)
𝑢=ℜ

(𝐿)
𝑣 =ℑ

3

𝐿)
𝑣 = ℑ(

4

(𝐿)
𝑢=ℜ

5

5
4
3
2

1

1

0

𝑢=1

𝑣=

0

−1 −3 −2 −1

0

1

𝐷5 > 1412

3−1 −3 −2 −1𝑥

𝐷5 < 1
0

1

2

3

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Vladi06.jpg
Fig. 14.8: 𝑢+i𝑣 = tai(𝑥+i𝑦) by (14.40) at 𝑁 = 51 and agreement 𝐷5 (𝑥+i𝑦)

As in figures 14.6, 14.7, and 14.8, the levels for the agreement are drownwith increment 2, beginning with 2; one additional level 𝐷5 = 1 is shownwith thick line. Inside the inner loop, the agreement with at least 14
digits takes place.
Approximation tai by (14.40) significantly extends the domain, where
the tetration can be precisely evaluated through elementary functions.
For positive values of ℑ(𝑧), tetration tet(𝑧) can be approximated withtet(𝑧) ≈ tai(𝑧)(14.42)For negative ℑ(𝑧), tertian can be approximated withtet(𝑧) ≈ tai(𝑧 * )*(14.43)

These representations are sufficient to plot map in figure 14.4d. I assume, that the transfer equation (14.1) is applied some integer number oftimes, in order use tai(𝑧) with the argument from the strip |ℜ(𝑧)| ≤ 1/2.
However, the expansions above do not provide the accurate approximation of tet(𝑧) at |ℑ(𝑧)| > 5.
One could extend the range of approximation, using the truncated Taylor expansions at point 5 i (or even 6 i), this would significantly extend
the range of approximation, and continue such an exercise with new andnew points along the imaginary axis. However, there exist more intelligent and elegant way to deal with cases, when the imaginary part of
the argument is large. This way is described in the next section.
195

8

Asymptotic expansion

The approximation of tet at large values of its argument can be build
up using the asymptotic representation

tet𝒜 (𝑧) = 𝐿 +∑︁𝒜𝑚,𝑛 exp (𝐿𝑛𝑧 + 𝛼𝑚𝑧)

(14.44)𝑛,𝑚𝐿 ≈ 0.31813150520476413+1.3372357014306895 i is, as before, the fixed
point of logarithm, 𝐿 = ln(𝐿), and 𝒜 are constant coefficients.
Substitution 𝑓 = tet𝒜 into the transfer equation (14.1) gives the chain ofequations for coefficients 𝒜. These equations do not determine 𝒜𝑚,0 ; so,the solution still has the countable set of “free” parameters for natural
𝑚. Difficulty of determination of these parameters had been discussed in1950 by Hellmuth Kneser [10]. However, even a relatively small amountof terms taken into account in expansion (14.44) can be used for the
precise approximation and evaluation of tetration at large values of theimaginary part of the argument.
Looking at the general (and a little bit ugly) expansion (14.44), I suggest
the approximate, but more beautiful formulafima(𝑧) =𝑁∑︁𝑎𝑛 𝜀𝑛 + 𝛽𝜀 exp(2𝜋i𝑧) ,(14.45)𝑛=0

where the small parameter

𝜀= exp(𝐿𝑧 +𝐿𝑟)

(14.46)

Mnemonics of name fima is following: Functional expansion for large
IMAginary part of the argument. Substitution of 𝑓 (𝑧) = fima(𝑧) +𝑂(𝜀𝑁 +1 ) into the transfer equation (14.1) gives the coefficients

𝑎0 =
𝑎1 =

𝐿 ≈ 0.31813150520 + 1.33723570143 i
1
1/2
𝑎2 =
≈ −0.1513148971 − 0.2967488367 i
𝐿−1
𝑎2 + 1/6
2+𝐿
𝑎3 ==≈ −0.036976+0.098730 i
𝐿2 − 16(𝐿−1)(𝐿2 −1)
6 + 6𝐿 + 5𝐿2 + 𝐿3
𝑎4 =≈ 0.02581−0.01738 i
24(𝐿−1)3 (𝐿+1)(𝐿2 +𝐿+1)
24+36𝐿+46𝐿2 +40𝐿3 +24𝐿4 +9𝐿5 +𝐿6
𝑎5 =120(𝐿−1)4 (𝐿+1)2 (1+𝐿 +2𝐿2 +𝐿3 +𝐿4 )≈ −0.0079444196+0.00057925018 i
196(14.47)(14.48)(14.49)(14.50)(14.51)(14.52)

(𝐿)
𝑣 =ℑ

(𝐿)
𝑢=ℜ

(𝐿)

(𝐿)
𝑣 =ℑ

𝑢=0
𝑢=0 𝑣 =1

𝑢=0 𝑣 =1

1
0

(𝐿)
𝑢=ℜ

𝑢=ℜ

(𝐿)

2

𝑣 =ℑ

3

(𝐿)
𝑣 =ℑ

4

(𝐿)
𝑢=ℜ

𝑦

−7 −6 −5 −4 −3 −2 −1

𝑦

0

1

2

3

4

5

6

7

𝑥

𝐷fifi > 14

4
3
2
1
0

𝐷fifi < 1
−7 −6 −5 −4 −3 −2 −1

0

1

2

3

4

5

6

7

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Vladi03.jpg
Fig. 14.9: Top:

𝑢+i𝑣 = fima(𝑥+i𝑦) by (14.45); bottom: map of 𝐷fifi = 𝐷fifi (𝑥+i𝑦)by (14.55)

It is not difficult to take into account more terms, but 𝑁 = 5 alreadyallows to cover the rest of the complex plane (not covered with approximations “maclo” and “tai”) with accurate approximations of natural
tetration.
Coefficients 𝑅 and 𝛽 in the right hand sides of formulas (14.45) and(14.46) remain as “adjusting parameters”. Their values are chosen inorder to approximate tetration, evaluated with a little bit slower Cauchy
integral:𝑟 ≈ 1.0779614375280 − 0.94654096394782 i
𝛽 ≈ 0.12233176 − 0.02366108 i(14.53)(14.54)

These values can be interpreted as approximations of the fundamental
mathematical constants. I suggest to call them “the Kneser constants”,
as the expansion with these coefficients had been suggested in 1950
by H.Kneser [10]. Many digits in approximations of these constants
can be calculated, in a way, similar to that in centuries 19 and 20 themathematicians competed in precision of evaluation of number 𝜋 .
Complex map of function fima is shown in the top picture of figure
14.9; the upper half of the complex plane is shown. This map should
be compared to the map of tetration in figure 14.4 (although the levels
197

ℜ(𝐿) and ℑ(𝐿) are not drawn in figure 14.4) and to maps of otherapproximations in figures 14.6, 14.7 and 14.8
There is no fundamental limit on the precision of evaluation of tetration(for example, through the Cauchy integral), so, parameters 𝛽 and 𝑟
in should be considered as fundamental mathematical constants. The
numerical computations, described in this book, have precision of orderof 14 decimal digits (that is close to the best precision achievable withvariable complex double), and parameter 𝑟 is evaluated with the similar
precision. Precision of evaluation of parameter 𝛽 is not so high; perhaps,
calculus with variables “long complex double” are necessary to improve
the precision of evaluation of 𝛽 and add more digits in the right handside of equation (14.54).
In order to show the residual at the substitution 𝑓 = fima into thetransfer equation (14.1), figure (14.9) shows the agreement(︂|fima(𝑧) − exp(fima(𝑧 −1))|
𝐷fifi (𝑧) = − lg|fima(𝑧)| + | exp(fima(𝑧 −1))|)︂(14.55)This agreement can be considered as an estimate, for how many orders of magnitude the value of the function is larger, than the errorof its evaluation with approximation fima. As in the previous maps ofagreement, the levels are drown with interval two orders of magnitude;
only for level 𝐷fifi = 1, the exception is done; this level is shown withthick line. Below this level, the approximation fima does not reproduce
even the qualitative behaviour of natural tetration. In the upper region,
above the highest level, contrary, the approximation provides at least
14 significant figures, that is close to the maximal precision, achievablewith variables complex double.
This section suggests the asymptotic approximation denoted with name“fima” by (14.45). Approximation fima is valid in the most of the upper
part of the complex plane. Its conjugation 𝑧 ↦→ fima(𝑧 * )* provides
the approximation for the most of the lower part of the complex plane.
With the transfer equation, these approximations can be extended also
to the larger values of the real part of the argument. Together withapproximations “maclo” and “tai”, the whole complex plane happens
to be covered with overlapping regions, and for each of these region,
the efficient approximation based on the series expansion, is described.
Now it would be methodically correct to analyse, verify the overlappings,
agreement of these approximations. This overlapping is considered in
the next section.
198

𝑦
5
4

𝑦
5

𝐷 > 14

4

3

3

2

2

1

1

0
−3 −2 −1

0

𝐷<1

1

2

𝐷 > 140−3 −2 −1

3 𝑥

0

1

2

3 𝑥

https://mizugadro.mydns.jp/t/index.php/File:Vladi07.jpg
Fig. 14.10: Comparison of approximations tai by (14.40) to fima by (14.45) and to
maclo by (14.36): agreements 𝐷 = 𝐷6 (𝑥+i𝑦) and 𝐷 = 𝐷7 (𝑥+i𝑦) by (14.56),(14.57)
in the complex z-plane.

9

Comparison of approximations

On the base of representation of natural tetration through the Cauchy
integral, the coefficients of various expansions of tetration are evaluatedand the approximations with elementary functions are suggested. In
this section, the mutual agreement or these representations is analysed.
The left hand side of figure 14.10 shows the agreement of approximation
tai by (14.40) with approximation fima by (14.45):(︂|tai(𝑧) − fima(𝑧)|𝐷6 (𝑧) = − ln|tai(𝑧)| + |fima(𝑧)|)︂(14.56)

The right hand side of figure 14.10 shows agreement of approximation
tai by (14.40) with approximation maclo by (14.36)"(︂|tai(𝑧) − maclo(𝑧)|𝐷7 (𝑧) = − ln|tai(𝑧)| + |maclo(𝑧)|)︂(14.57)

Figure 14.10 indicates, how to choose the appropriate approximationof tetration dependently on the imaginary part of the argument 𝑧 atmoderated values of |ℜ(𝑧)| < 1. The boundary between the domains of
the approximations should go through the loops, where 𝐷 > 14. While|ℑ(𝑧)| ≤ 1.5, let approximation maclo be used; At 1.5 < ℑ(𝑧) ≤ 4.5,
let the approximation tai be used, and, at even larger values, let theevaluation be performed with approximation fima.

199

ℑ(𝑧)

ℑ(𝑧)

𝐷 > 145

12 < 𝐷 < 14

5

4

4

3

3

2

2

1
0

−2

0

2

𝐷<1

1

ℜ(𝑧)

0

𝐷 > 14

𝐷<1
−2

0

2

ℜ(𝑧)

https://mizugadro.mydns.jp/t/index.php/File:Vladi08.jpg
Fig. 14.11: Agreement 𝐷 = 𝐷8 by (14.59), left; the similar agreement for the contour
integral with base domain shifted for −0.5.

Looking at figure 14.10, I suggest the following approximation:⎧⎪fima(𝑧)⎪⎪⎪⎪⎪⎨ tai(𝑧)fse(𝑧) =maclo(𝑧)⎪⎪⎪tai(𝑧 * )*⎪⎪⎪⎩fima(𝑧 * )*, 4.5 < ℑ(𝑧), 1.5 < ℑ(𝑧) ≤ 4.5, −1.5 ≤ ℑ(𝑧) ≤ 1.5, −4.5 ≤ ℑ(𝑧) < −1.5

(14.58)ℑ(𝑧) < −4.5,

This approximation can be compared to previous results. The left hand
picture of figure 14.11 shows the agreement(︂)︂|fse(𝑧) − 𝐹4 (𝑧)|(14.59)𝐷8 (𝑧) = − lg|fse(𝑧)| + |𝐹4 (𝑧)|of approximation fse with the approximation 𝐹4 obtained through thedirect implementation of the contour integral.
Figure 14.11 reveals the defects of each approximation. The jumps atℑ(𝑧) = 1.5 and at ℑ(𝑧) = 2.5 should be attributed to the transition fromfunction maclo to function tai and from function tai to function fimain the combination FSE. Jumps at half-integer values of ℜ(𝑧) should
be attributed to the discontinuities of function 𝐹4 , which extends theapproximation with the contour integral, valid for |ℜ(𝑧)| < 1, from theinterval |ℜ(𝑧)| ≤ 1/2. The rounding errors appear as irregular dots.
Within the strip |ℜ(𝑧)| < 1.5, the irregularities of all three approximations are of order of 10−14 .
The goal is to cover with efficient (fast and accurate) approximations atleast the strip ℜ(𝑧) ≤ 0.5; then, values of natural tetration for the whole200

complex plane can be expressed through the transfer equation (14.1) inthe right hand side of the complex plane, and through the ‘inverted”
equationln(tet(𝑧)) = tet(𝑧 −1)(14.60)in the right hand side. The left map in figure (14.11) indicates, that thegoal is achieved; agreement with approximately 14 decimal digits takes
place in significantly wider part of the complex plane. The approximations above are used for the fast implementation.
After to see the agreement discussed above, I had declared, that sincenow, the natural tetration can be evaluated so fast and so precisely, as
other special functions, known since century 20. Then Henryk Trappmann asked me to make one additional numerical test. He wanted tosee, if the same tetration can be evaluated, if I misplpace the contour
Ω in the Cauchy integral [105], moving it to the right. I recognised thisas a trap (which would correspond to the last name of Henryk): if we
displace the contour to the right, the derivatives of tetration becomeslarger, and, with the same algorithm, we get lower precision. But I
agreed to displace the contour for 1/2 to the left.
With the displaced contour, the same ab initio evaluation of tetration
had been performed. Tetration tet(−1/2+𝑖𝑦) for real 𝑦 had been evaluated; then, with the Cauchy integral and equations (14.1),(14.60), theapproximation had been extended to the whole complex plane, in thesimilar way as with the first algorithm of evaluation of tetration [54].
The result is compared to the approximations with expansions in theway, similar to that of by (14.59); the new approximation is used instead of 𝐹4 . The resulting agreement is shown in the right hand side
map in figure (14.11).
Figure 14.11 reveals defects of approximations mentioned above. The
discontinuities in formula (14.58) are seen with horizontal jumps alonglines ℑ(𝑧) = 1.5 and ℑ(𝑧) = 4.5.Discontinuities of the initial, “primary”
approximation appear with the vertical jumps along half-integer valuesof ℜ(𝑧). The similar discontinuities are seen also for the evaluation
with displaced contour at integer values of ℜ(𝑧). All these jumps of thecompared approximations are at the level of 10−14 , and this confirms
the declared estimate of the precision of the evaluation of the natural
tetration.
The agreement in the right hand side of figure 14.11 happened to be201

even better, than that in the left hand side. Henryk had been satisfied
with that test. For the natural tetration, the new contour of integration
happened to be a little bit better, than the initial choice. In such a way,
the analogy with lazy engineer (or with stupid selfish tyrant), mentioned
above, gets the confirmation: the initial contour of integration in theoriginal publication [54] is not best. However, I still think, that thesimplicity of that contour and the good agreement (figure 14.11) should
be considered as some kind of excuse for the voluntaristic choice of thecontour.

10

Implementation

After the tests, described in the previous section, for the numerical implementation, the following approximation is used: tet(𝑧) ≈ FSE(𝑧),with⎧
FIMA(𝑧) , 4.5 < ℑ(𝑧)
⎪
⎪
⎪
⎪
⎪
⎪
, 1.5 < ℑ(𝑧) ≤ 4.5⎨ TAI(𝑧)FSE(𝑧) =MACLO(𝑧) , −1.5 ≤ ℑ(𝑧) ≤ 1.5⎪⎪⎪
TAI(𝑧 * )* , −4.5 ≤ ℑ(𝑧) < −1.5⎪⎪⎪⎩
FIMA(𝑧 * )* ,ℑ(𝑧) < −4.5

(14.61)where{︃FIMA =

fima(𝑧), ℑ(𝑧) > 4+0.2379 ℜ(𝑧)exp(FIMA(𝑧 −1)), ℑ(𝑧) ≤ 4+0.2379 ℜ(𝑧)⎧⎪⎨tai(𝑧), |ℜ(𝑧)| ≤ 0.5
TAI =log(TAI(𝑧 +1)) , ℜ(𝑧) < −0.5⎪⎩
exp(TAI(𝑧 −1)) , ℜ(𝑧) > 0.5⎧⎪
tai(𝑧), |ℜ(𝑧)| ≤ 0.5⎨
MACLO =
log(MACLO(𝑧 +1)) , ℜ(𝑧) < −0.5⎪⎩
exp(MACLO(𝑧 −1)), ℜ(𝑧) > 0.5(14.62)(14.63)(14.64)This approximation provides of order of 14 correct significant figuresof the holomorphic tetration tet and agrees with the previous results[54]. Up to my knowledge, up to year 2016, function FSE above is themost precise and the fastest among ever reported approximations of thetetrational. However, in 2016-2017, the new results by William Paulsen,
Samuel Cowgill appear; they are not included here. In such a way, thenumerical implementation of tetration can be improved.
202

𝑦
𝑣=

1.4 𝑢 = 0.2

0.

4

3
1.4 𝑢 = 0.2

𝑣 =1

𝑢=0

1

𝑣 =0

0 cut
−1

𝑣 =0

𝑣 = −1

−3

−2

(𝐿)

0 .2

−1

𝑣 =ℑ *
(𝐿 )

−4
−4

𝑢=

𝑢=ℜ

𝑢=ℜ
(𝐿)

𝑣

1.4
=−

𝑣 =ℑ *
(𝐿 )

−2
−3

𝑢=0

𝑢=

𝑣=

2

0

𝑢=

0.4
−1.4
=
𝑣

1

2

3

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Tetma.jpg
Fig. 14.12: 𝑢+i𝑣 = tet(𝑥+i𝑦)

Mnemonics of the name FSE is obvious: Fast Super Exponent. The
C++ implementation of this algorithm is loaded as https://mizugadro.
mydns.jp/t/index.php/Fsexp.cin ; this approximation is used to plot
the detailed map of tetration in figure 14.12, used at cover of this Book.
Many terms are kept in approximations (14.40) and (14.36) in order toprovide the wide range of the overlapping in figures 14.10 and 14.11.
At the final step of the implementation, the number of terms can be
reduced, boosting the algorithm, without loss of the precision. In particular, this applies to the evaluation of tetration along the real axis: it
is sufficient to approximate tet(𝑧) for |𝑧| ≤ 1/2, using only a quarter ofthe radius of the precise approximation with function maclo.
For iterates of the exponent, the inverse function, id est, arctetration,or abelexponent, is also required. It is considered in the next chapter.
203

Chapter 15
Natural arctetration𝑢 = 2.2

𝑦
4
3
2
1
0
−1
−2
−3
−4
−5

𝑢=2

𝑣 = 0.6𝑣 = 0.4

𝑣 =0

−6

−4

−2

0

2

4

https://mizugadro.mydns.jp/t/index.php/File:Vladi02.jpg

6𝑥Fig. 15.1: 𝑢+i𝑣 = ate(𝑥+i𝑦)

The inverse function of tetration, id est, arctetration, or abelexponent,
is denoted with name ate; ate = tet−1 . Complex map of arctetration isshown in figure 15.1.
Arctetration satisfies the Abel equation(︁)︁
ate exp(𝑧) = ate(𝑧) + 1(15.1)and the additional conditionate(1) = 0(15.2)

Properties of functions ate and the algorithm for the evaluation are
described in this chapter.
204

1

Evaluation of arctetration

Arctetration can be evaluated as inverse function of tetration, using the
Newton method. Function ate(𝑧) appears as limit of sequence 𝑔𝑛 withthe recurrent relation

tet(𝑔𝑛 ) − 𝑧(15.3)
tet′ (𝑔𝑛 )The derivative of tetration can be approximated, differentiating the approximations of tetration with elementry functions described in the previous section. The representation through the Cauchy integral [54] also
allows the straightforward differentiation. However, in this case, several
iterates by (15.3) are required to evaluate the arctetration.
𝑔𝑛+1 = 𝑔𝑛 +Evaluation of arctetration through tetration using equation (15.3) issignificantly slower, than evaluation of tetration. In addition, the initialapproximation 𝑔0 should be specified. This specification should carryabout the cutlines. In Figure 15.1, these cut lines are drawn parallelto the real axis. Over-vice, the recurrence by (15.3) returns a value
from any of branches of the corresponding multivalued function, andthe question about the range of holomorphism becomes difficult.
In order to get efficient approximation for the arctetration ate, I deal
with the corresponding Abel equation (15.1), rather than with recurrences by (15.3). It worth to approximate arctetration with some function, which reproduce at least the leading terms of the asymptotic expansion of ate. This approximation is constructed below.
Arctetration, as solution of the Abel equation (15.1), should have singularities in the fixed points of logarithm 𝐿 and 𝐿* . From the precious
chapter, we already know, that the dominant term of the asymptotic
expansion appears as the fixed point plus the appropriate exponential.
This indicates, that the expansion of arctetration should begin withlogarithm. The efficient approximation of arctetration can be obtained
through the expansion of function ℎ byln(𝑧 −𝐿) ln(𝑧 −𝐿* )ℎ(𝑧) = ate(𝑧) −−(15.4)𝐿
𝐿*
Function ℎ can be expanded to the Taylor series at unity. This expansion
leads to the approximation
𝑁 −1ln(𝑧 −𝐿) ln(𝑧 −𝐿* ) ∑︁++𝑢𝑛 (𝑧 −1)𝑛fsl(𝑧) =*𝐿𝐿
𝑛=0(15.5)

ate(𝑧) = fsl(𝑧) + 𝑂(𝑧 −1)𝑁

(15.6)

205

𝑦

𝑦

1

1

0

0

−1

−1

−2

−1

0

1

2 𝑥

−2

𝐷A < 1

15

𝑦

1
0

15

𝐷B < 1

−1
−1

0

1

2 𝑥

−2

−1

https://mizugadro.mydns.jp/t/index.php/File:Vladi10.jpg

0

1

2 𝑥

Fig. 15.2: 𝑢+i𝑣 = slo(𝑥+i𝑦) by (15.5), left, and the agreements by 𝐷A = 𝐷A (𝑥+i𝑦),

𝐷B = 𝐷B (𝑥+i𝑦), by formulas (15.7), (15.8)

Approximations of the first 30 coefficients of this expansion are shownin table 15.1. Complex map of function fsl by (15.5) at 𝑁 = 70 is shownin the left hand side picture of figure 15.2 in the same notations, as
in figure 15.1. The central part of left map in figure 15.2 looks as a
fragment from figure 15.1.
Table 15.1:

𝑛
0
1
2
3
4
5
6
7
8
9

𝑢𝑛
1.41922521550451−0.02606629029752
0.00173304781808−0.00001952130725−0.00006307006450
0.00002567895998−0.00000559010027−0.00000007279712
0.00000065148872−0.00000027698138

Coefficients 𝑢𝑛 in expansion (15.5).
𝑛
10
11
12
13
14
15
16
17
18
19

𝑢𝑛
0.00000003111805
0.00000002940887−0.00000001896929
0.00000000351784
0.00000000204270−0.00000000171995
0.00000000039882
0.00000000019328−0.00000000019113
0.00000000004947

𝑛
20
21
22
23
24
25
26
27
28
29

𝑢𝑛
0.00000000002293−0.00000000002462
0.00000000000666
0.00000000000322−0.00000000000354
0.00000000000096
0.00000000000051−0.00000000000055
0.00000000000014
0.00000000000009

Range of validity of approximation 15.5 is limited. In order to show
this range, the central and right hand side maps in figure 15.2 show theagreements(︂)︂|fsl(exp(𝑧))−1 − fsl(𝑧)|𝐷A (𝑧) = − lg
|fsl(exp(𝑧))−1| + |fsl(𝑧)|(︂)︂|fsl(ln(𝑧))+1 − fsl(𝑧)|𝐷B (𝑧) = − lg
|fsl(ln(𝑧))+1| + |fsl(𝑧)|(15.7)(15.8)

Inside the inner loops in the central and right hand side pictures of figure 15.2, the agreement is better than 14. These domains are marked
206with symbol “15”. Figure 15.2 indicates, that at |𝑧−1| < 1.4, the approximation fsl(𝑧) provides of order of 14 significant figures. The precisionof this approximation is a little bit worse in vicinity of fixed points 𝐿
and 𝐿* . This could be expected: First, there, the small variation ofargument leads to the significant variation of value of function. Second,
these points are at the edge of the range of convergence of expansion inequation (15.5).
For implementation of natural arctetration, it is sufficient to prepare theefficient primary approximation for the domain sickle, defined withsickle = {𝑧 ∈ C : ℜ(𝑧) ≥ 𝐿, |𝑧| < 𝐿}

(15.9)

then, for other values of the argument, function can be expressed throughthe Abel equation (15.1). The region sickle is shaded in maps of figures
15.1, 15.2. This leads to implementation, denoted with FSL. I check theself-consistency of implementation FSL with implementation FSE by(14.61). The numerical test of the relations below had been performed:

ate(tet(𝑧)) = 𝑧
tet(ate(𝑧)) = 𝑧(15.10)(15.11)

These relations are tested for the complex double implementations tet ≈FSE and ate ≈ FSL. Figure 15.3 shows the maps of the agreements(︂)︂
|FSL(FSE(𝑧)) − 𝑧|
𝐷at (𝑧) = − ln
|FSL(FSE(𝑧))| + |𝑧|)︂(︂|FSE(FSL(𝑧)) − 𝑧|
𝐷ta (𝑧) = − ln
|FSE(FSL(𝑧))| + |𝑧|(15.12)(15.13)In figure 15.3, the left hand side map shows 𝐷 = 𝐷at (𝑥+i𝑦) and theright hand side map shows 𝐷 = 𝐷ta (𝑥+i𝑦) in the 𝑥, 𝑦 planes. The levels
𝐷 = const are drawn with interval 2; as in other maps of agreement,
the additional level 𝐷 = 1 is shown with thick line. This line indicatesthe boundary of the range of validity of relation (15.10).
As for many other inverse functions, the range of validity to relation(15.10) is limited. Width of the strip, where the relation (15.10) takes
place, is determined by the asymptotic periodicity of tetration in theupper and the lower parts of the complex plane. In vicinity of the realaxis, the strip becomes wider, showing some kind of along the real axis
to infinity. Thickness of this “beak” reduces quickly at the growth of thereal part of the argument.
207

𝑦

𝑦

𝐷<1

4

4

10 < 𝐷 < 13

4

4

3

3

12 < 𝐷 < 14

2
1

2

𝐷<1

0

1

𝐷 > 14−1−2−2

12 < 𝐷 < 14

−4

−4
−5

𝐷<1
−3 −2 −10

1

12 < 𝐷 < 14

−3

10 < 𝐷 < 12

−5

𝐷 > 140−1−3

12 < 𝐷 < 14

2

3

𝑥

−3 −2 −10

https://mizugadro.mydns.jp/t/index.php/File:Vladi11.jpg
Fig. 15.3:

1

2

3

𝑥

𝐷 = 𝐷at (𝑥+i𝑦), left, and 𝐷 = 𝐷ta (𝑥+i𝑦), right, by (15.12) and(15.13)The numerical tests confirm, that the approximations of tetration andarctetration are self-consistent. The complex double implementations
provide of order of 14 significant figures.

2

About names

The numerical implementations of algorithm FSE and FSL describedabove are loaded to TORI as

https://mizugadro.mydns.jp/t/index.php/Fsexp.cin andhttps://mizugadro.mydns.jp/t/index.php/Fslog.cin
The names of these routines are discussed in this section.
Names of function FSEXP and FSLOG are historic. They may mean
“Fast Super EXPonential” and “Fast Super LOGarithm. Hernryk Trappmann even wanted to add my last name to the identifier of each of these
two functions. He had believed, that “my tetration” is not unique, andnot so principal, as I claim, using names “tet” and “ate”. Henryk wanted

208

to use names “tet” and “ate” for the “true” tetration and arctetration,“more natural”, more “true”, than the functions I had constricted.
After the long discussion and the heavy deduction, Henryk had to agree,that the functions I have constructed are unique [74], and, in this sense,true and the only true tetration and arctetration. The routines FSEXPand FSLOG were already implemented that time, and I decided not
to change the notations: the poor sistem of notation is still better andcauses less confusions, than two “good” systems of notations.
I need to mention, that name FSLOG (Fast Super LOGarithm) is evenmore idiotic, than FSEXP. Name FSLOG makes impression, that it issuperfunction of logarithm, while it is not really so. Superfunction ofnatural logarithm van be written as𝑧 ↦→ tet(−𝑧)(15.14)

I think, this function does not deserve to have a special name assigned.
I am far from eugenic ideas to refine the human rase, nor the system
of notations. The best system of notations should win the competition
with other notations. All this should be considered as my excuse to keepnames FSEXP and FSLOG for the approximations and the numerical
implementations of tetration tet and arctetration ate.
Many superfunctions of natural exponent can be constructed with transformation (2.17), just misplacing the argument of tetration with someperiodic real-holomorphic function. The range of holomorphism of thesetransforms is narrower, than the range of holomorphism of tetration.
Now, I see no need to give them special names.
I expect, in future, even more efficient implementations for tetration will
be suggested. Then, they may be called with the same names, as thename of the functions tet and ate, in the same way, as in the algorithmic
languages the implementation of sin is denoted with the same name as
the function.
After to eliminate the potential confusion with names, the tetration andarctetration can be used for the iterates of the exponent. These iterates
are described in the next section.
209

3
2

𝑛=2
−2

−1

00

𝑥

(𝑥)
n
l
𝑦=

1
1

2

3

𝑥

−1

−3

𝑛 = −3𝑛 = −3.1
𝑛 = −3.2

0.1
𝑛 = −0.2
𝑛 = −0.3

−2

𝑛=−

𝑛=

0

𝑛 = 1 −3𝑛 = 0.6
𝑛 = 0.4
𝑛 = 0.3
𝑛 = 0 .2
𝑛 = 0.1𝑦=𝑦=𝑛 = 3.2
𝑛 = 3.1
𝑛=3
𝑛 = 2.9

exp
(𝑥)

𝑦

https://mizugadro.mydns.jp/t/index.php/File:ExpIte4T.jpg
Fig. 15.4:

3

𝑦 = exp𝑛 (𝑥) by (15.15) for varuous 𝑛

Iterates of exponent

Tetration tet and arctetration ate, as superfunction and abelfunction of
exponent, specify, determine the non-integer iterates:

exp𝑛 (𝑧) = tet(𝑛 + ate(𝑧))(15.15)Here, number 𝑛 of iterates, has no need to be integer (although, ofcourse, can be integer too). For real values of argument, iterates of
exponent by equation (15.15) are shown in figure 15.4, 𝑦 = exp𝑛 (𝑥).
Lines, that correspond to integer 𝑛 (except 𝑛 = 0), are thick. Theselines correspond to 𝑦 = exp(exp(exp(𝑥))), 𝑦 = exp(exp(𝑥)), 𝑦 = exp(𝑥),𝑦 = ln(𝑥), 𝑦 = ln(ln(𝑥)), 𝑦 = ln(ln(ln(𝑥))). Higher integer iterate happen
to be out of range of the figure.
Complex maps of iterates of the exponent are collected in figure 15.5.

210𝑢 = −1
𝑢=
𝑢=1
2

𝑣
𝑣==1
0.
8

𝑢 = 1 .4
𝑢 = 1.2
1

1
𝑣=

𝑢=

0

2

𝑢=

𝑢=
𝑢 1
𝑢 == 2
3

−1
𝑢=
0
𝑢=
1
𝑢=2

7
1.3
3
𝑣≈

0 .2

0.4

𝑢=
𝑢= 1
2

𝑣 =0

𝑢=

𝑣=

1

cut

𝑣 =0
𝑣 =3
𝑣 =2

𝑣 ≈ 1.337 𝑣 = 1

cut−3 −2 −1 0
12
3𝑥
−3 −2 −1 0
1
https://mizugadro.mydns.jp/t/index.php/File:Expitemap.jpg

0Fig. 15.5: Maps of iterates of natural exponent by (15.16)

211

𝑢=2

𝑛 = −0.1𝑣 =2𝑣 ≈ 1.337 𝑣 = 1

𝑢=1

cut

𝑣 ≈1
. 337

𝑣 =4
𝑣 =3
𝑣 =2
𝑣 =1

1

1

𝑣 = 5 𝑛 = −0.2
𝑣 =4
𝑣 =3
𝑣 = 2 cut𝑣 =1
𝑣 = 0 cut

1

2

𝑣 =0

𝑢=0

𝑛 = 0.1

𝑣 =1

cut

𝑢=−

3

2

1

𝑢=

0
𝑦

7

𝑣=

cut

0

1

cut

1.3
3

𝑢=0

−1

2

𝑢=

𝑢 ≈ 0.318

𝑢=𝑛 = 0.2𝑣≈

𝑛 = −0.5
𝑣 =3
𝑣 =1

1

0 .6
=
𝑣 0 .4𝑣 = .2
𝑣 =0

0
=
𝑢

𝑢=

3

cut

𝑣=

𝑢=2

0
𝑦

cut

𝑢=

1

cut

𝑢 = −21
−
𝑢= 0
𝑢=

𝑢=0
𝑢=1
𝑢=2

−
0.
2

𝑛 = 0.5

2

𝑢

𝑢=0

=1

2

3

0

𝑢=

0
𝑦

cut

𝑛 = −0.8

cut

𝑢=

1

𝑢=

0

𝑣=

0 .6
=
𝑣 0 .4𝑣= 2
𝑣 = 0.
𝑣 =0

cut

𝑛 = 0.8

2

1

1

cut

1

3

𝑛 = −0.9
𝑢=

𝑢=0
1
=
𝑢

0

0
𝑦

0
𝑢=

1

cut

𝑣=

=
2 𝑣

0 .6
=
𝑣 = 0 .4
𝑣 0 .2
𝑣=
𝑣 =0

𝑣 =3

𝑛 = 0.9

𝑢=

3

𝑣 =0

𝑢=

1
0
𝑦

𝑛 = −1
𝑢=

𝑢=0

𝑢=−

2

𝑢 = −0.2

3

𝑛𝑣 =
1
=0

0 .2

𝑦

2

3

𝑥

Twelve maps are shown for

𝑢+i𝑣 = exp𝑛 (𝑥+i𝑦)(15.16)

with lines 𝑢 = const and lines 𝑣 = const in the 𝑥, 𝑦 plane for various
values of the number 𝑛 of iterate; this 𝑛 is printed with big font in theupper left corner of each map. The maps are symmetric with respect
to reflection from the abscissa axis (the only, the imaginary part of thefunction changes its sign). So, the only upper part of the complex plane
is presented in each map.
Maps at the top of figure 15.5 correspond to 𝑛 = 1 and 𝑛 = −1; these arecomplex maps of exponent and of logarithm. First of them reproduces
part of figure 14.2. The exponent is holomorphic in the whole complex
plane, but the logarithm has branch point at zero and the cut along thenegative part of the real axis.
The second and following rows of the figure represent the non-integer
iterates. These iterates have two additional cuts along lines 𝑦 = ±ℑ(𝐿);here 𝐿 ≈ 0.3181315+1.3372357 i is fixed point of logarithm, id est, solution of equation 𝐿 = ln(𝐿). By default, all the cut lines are directed
parallel to the real axis (axis 𝑥 in the figure, abscissa) toward the negative direction of the real axis. In such a way, for negative non-integer𝑛, the map has 3 cut lines (and that in the lower half-plane is not seen,
as it is out of field of view of the map).
The thick lines in figure 15.5 corresponds to the integer values of 𝑢 or 𝑣 .
The thin lines are drawn with interval 0.2; the additional lines 𝑢 = ℜ(𝐿)and 𝑣 = ℑ(𝐿) are also drawn. These lines always cross each other at the
fixed point 𝐿.
Figure 15.5 shows the gradual transition of the map for the exponential(top of the left column) the map for the logarithm (top of the right handside column). As the number 𝑛 of iterate reduces from unity to zero,
the web of the lines 𝑢 = const and lines 𝑣 = const rotate around the fixed
point 𝐿, and become uniform rectangular grid at 𝑛 = 0. At this value,
the horizontal cuts along lines ±ℑ(𝐿) disappear, but they appear again,as 𝑛 becomes negative non-integer. At 𝑛 < 0, the additional branch pointcomes from −∞ at the real axis and moves toward zero, as 𝑛 becomes
minus unity. With integer 𝑛, the branch points 𝐿 and 𝐿* disappear.
Maps of non-integer iterates can be plotted also for other transfer functions, considered in this Book. The readers are invited to download theimplementations of the superfunctions and the abelfunctions, and plot
the corresponding complex maps of the iterates.
212

4

Lessons of natural tetration and arctetration

On the base of representation of tetration through the Cauchy integral,through the solution of the integral equation (14.22), one can express
also derivatives of this function; differentiation of the integrands in theright hand side of (14.22) is straightforward. Namely in this way, thederivatives of tetration for real and for pure imaginary values of theargument had been evaluated for the tables 1 and 2 in publication [54].
In this section, I suggest some philosophic speculations about tetration
and arctetration.
Complex map of tetration tet is shown in figure 14.12, and its behaviour
along the real axis is shown in figure 14.1. Properties (14.16)-(14.20) first
were observed with various approximations of tetration with elementatyfunctions, and then postulated. The approximations reproduce valuesof tetration in vicinity of integer values of the argument,tet(−2 + 𝜀) = log(𝜀) + const + 𝑂(𝜀)tet(−1) = 0
tet(0) = 1
tet(1) = etet(2) = e2

(15.17)(15.18)(15.19)(15.20)(15.21)

Then, the agreement at the substitution of the fitting function into thetransfer equation (14.1) had been minimised for complex values of theargument.
The behaviour similar to properties (14.16)-(14.20) had been detectedwith various fitting functions. Then these properties were formulated
as definition of tetration, id est, just postulated. First, I did not expect
this set of postulates to be self-consistent. Contrary, I tried to findsome contradiction; I expected to use such a contradiction as a proof of
non-existence of holomorphic tetration. Such a non-existence would bean upgrade of the proof by M.Hooshmand [49], that uses the doubtful
assumption about monotonous behaviour of the derivative of tetration;
I tried to find a proof, that does not use this assumption. Expressionof tetration through the Cauchy integral [54] allows to make the precise
approximations [64], and no internal contradictions in the assumptions(14.16)-(14.20) had been detected. This leads to the conjecture about
existence and uniqueness of tetration, that later had been confirmedwith the careful analysis [74]. I show the first primitive approximations
in figure 14.4, as they answer the frequent question by colleagues: “ How213

did you guess?”.

I think this heuristic approach can be used also for
analysis of other (and more complicated) functional equations.
Tetration and arctetration significantly extend the arsenal of functions,available for the description of physical phenomena. In particular, thenon-integer iterates of exponent can be useful in description of processes,that grow faster than any polynomial, but slower than any exponent.
Following the TORI axioms, I formulate mainly the practical problems.
From the point of view of applications, not the proof by itself is important, but the strong indication, that the system of postulates is not
self-contradictory. The multiple (failed) attempts to reject the conjecture of existence and uniqueness can be considered as such indication.
Some “pure mathematicians” believe, that the only rigorous proof has
a scientific value. In order to show, that actually it is not so, I suggest
the example with the Euclid axioms of planimetry. Those axioms canbe deduced from the properties of the coordinate plane. It is not so
difficult, although first, one had to provide the accurate definitions of sin
and cos as solution of the corresponding system of differential equations,check that their properties lead to the Pythagoras theorem and otherproperties, known as the Euclid Axioms. In the elementary school,
however, till now, the teachers begin with the postulating the Euclid
axioms. I believe, the superfunctions should become a pretty elementary
tool, and their properties (including those of tetration) could be justpostulated - in the similar way, as the Euclid axioms. If someone wants
to reduce the amount of axioms, one may begin wight he Euclud axioms,
having no need to deal with tetration and other superfunctions. I hope,
one day, the beautiful, short, simple and rigorous proof of the existence
and uniqueness will be formulated.
Form my side, I make all possible efforts in order to simplify refutationof my concept (for the case, if one day someone will be able to refutethem). I load the figures from this book to my site ashttps://mizugadro.mydns.jp/t/index.php/Category:BookPlot
https://mizugadro.mydns.jp/t/index.php/Category:BookMap

and I supply them with generators in C++ and Latex. Everybody can
reproduce the figures, and plot the new figures, trying to find a hint to
any internal contradictions in the concepts suggested. Of course, anyother alternative hypothesis can be considered too, as it is shown infigure 15.6.
After to see, how the natural tetration comes from the Cauchy integral
in a pretty natural way, I had constructed similar maps for other values
214https://mizugadro.mydns.jp/t/index.php/File:BlackSheep.png

Two mathematicians go to the First International Congresson Superfunctions, and discuss the color of the ship they see from thetrain: Your assumption, dear colleague, seems to be not obvious, it isFig. 15.6:

not supported with observations.

Yet, all what we can conclude, that

there is at least one sheep in this country, and at least the right handside of this sheep is black.

of base 𝑏, namely, for 𝑏 = 10, 𝑏 = 2 and 𝑏 = 1.5, but I did not revealed any
new property, that could be difficult to expect, looking at the natural
tetration. The most of curves in figure 17.1, considered later, can beplotted with the Cauchy integral by very similar algorithms. I was sure,that my mission about tetration is finished. Then, Henryk Trappmannwanted still to reduce 𝑏; he asked√me, Wether I can evaluate in thesimilar way tetration to base 𝑏 = 2. I had to confess, that I cannot.
But I told, that I can do it by another way [61]. That “another way”
happened to be even simpler, than application of the Cauchi integral; so,
I described it in the previous√chapters as “regular iteration”. However,namely tetration to base 𝑏 = 2 is not described above; this tetration ismatter of the next chapter.

215

Chapter 16√

Tetration to base 𝑏 = 22

𝑦 = tet 2(𝑥)
√

1
−2 −1
−1

1

3

𝑥

𝑦=

−2

2

𝑥
−

−3

https://mizugadro.mydns.jp/t/index.php/File:Sqrt27t.jpg
Fig. 16.1:

𝑦 = tet√2 (𝑥)In chapter 13 above, the natural tetration is constructed and evaluated.
I mean, tetration to base 𝑏 = e ≈ 2.71 . For other bases, the definition oftetration should be generalised. This generalisation is suggested in thischapter. I try to follow the principles “from simple to complicated” and√“from specific to general”. First, I consider the specific base 𝑏 = 2. For
this base, the graphic of tetration is shown in figure 16.1. Namely forthis base, the graphic looks especially symmetric. Below I show, that
this is just visual impression, and the apparent symmetry 𝑥 ↔ −𝑦 isonly approximation.
216

1

Definition

Tetration to the real base 𝑏 > 1 is real-holomorhpic function 𝑓 = tetb ,that satisfies the transfer equation(16.1)𝑓 (𝑧 +1) = 𝑏𝑓 (𝑧)at least for ℜ(𝑧) > −2, and is bounded at least in the range |ℑ(𝑧)| ≤ 1,
and, in addition, the specific (the same for all 𝑏) value at zero is assumed:(16.2)

𝑓 (0) = 1

Here, function 𝑇 = exp𝑏 appears as the transfer function, and tetration
𝑓 as its superfunction.
The Reader is invited to check, that the natural tetration tet = tete ,considered in chapter 14, also falls into into this definition. Below, thetetration to various bases is considered. In particular, this chapter deals√
with the special case 𝑏 = 2. Exponential to this base is shown infigures 9.1 and 9.2. Namely for this case, in section 9, the growing super
exponential SuExp√2,5 , is constructed; graphic 𝑦 = SuExp√2,5 is shownin figure 9.4.

√

Here, for the same base 𝑏 = 2, id est, for the same transfer function,
I describe another superexponent, namely, tetration. Its explicit plotis shown in figure 16.1 and it is pretty different from the plot of thegrowing superexponent 𝐹 = SuExp√2,5 shown in figure 9.4.

2

Again regular iteration

In this section, again I use the regular iteration, as in chapter 9, in otherto construct another superfunction, namely, tetration, for the exponen√tial to base 2 as the transfer function. This construction appears as
an example to evaluation of tetration tet𝑏 for 1 < 𝑏 < exp(1/e).

√

In such a way, here I consider the case 𝑏 = 2. Explicit plot of thistetration is shown in figure 16.1. The construction below is quite analogous to the construction of the growing super exponent to the same
base, presented in chapter 9. Some formulas in this section are takenfrom publication [61].

√The fixed points of exponent to base 𝑏 = 2 are considered in chapter
9, see figure 9.1. These fixed points are 2 and 4. In that chapter,217

the superexponent 𝐹5 = SuExp√2,5 is built at the fixed point 𝐿 = 4.
Along the real axis, that superexponential grows monotonously from 4
to infinity. For that superexponent, the equation 𝐹5 (𝑧) = 1 has no real
solution; that superexponent is not tetration. In order to get tetration,
we should use the fixed point 𝐿 = 2. This case is considered in thischapter. The corresponding superfunction Φ can be expanded as follows:𝑓 (𝑧) = 2 + 𝜀 +

𝑀
−1
∑︁

𝑣𝑚 𝜀𝑚

(16.3)

𝑚=2

Φ(𝑧) = 𝑓 (𝑧) + 𝑂(𝜀𝑛 )(16.4)

𝜀 = exp(𝑘𝑧)(16.5)

where

while increment 𝑘 and coefficients 𝑣 are constants. Substitution of theasymptotic expansion 𝐹 = Φ in to the transfer equaiton)︁(︁√ )︁
2 𝐹 (𝑧)
𝐹 (𝑧 +1) = exp ln(︁

(16.6)

𝑘 = ln(ln(2)) ≈ −0.3665129205816643(16.7)

determines the increment

and leads to the chain of equations for coefficients 𝑣 . I set 𝑣1 = 1; then,ln(2)/4≈ −0.56472283831773236365
ln(2)−12ln(2) (2+ln(2))/24
𝑣3 = (ln(2)−1)(ln(2)0.33817758685118329988
2 −1) ≈

𝑣2 =(16.8)

Approximate values of coefficients 𝑣 are collected in table 16.1.
At fixed number 𝑀 of terms in the right hand side of equation (16.3),
function 𝑓 and be considered as approximation of superfunction withcertain asymptotics, namely, that grows at infinity, approaching the
fixed point 2. This superfunction appears as limit

Φ(𝑧) = lim 𝑇 −𝑛 (𝑓 (𝑧 +𝑛)) = lim log𝑏𝑛 (𝑓 (𝑧 +𝑛))
𝑛→∞

𝑛→∞

(16.9)

does not depend on the number 𝑀 of terms in the right hand side ofequation (16.3). However, at large 𝑀 , the limit in the right hand sideof (16.9) converges faster.

√

Tetration to base 𝑏 = 2 that satisfies condition (16.2), appears asfunction Φ with displaced argument,tet√2 (𝑧) = Φ(𝑥1 +𝑧)
218(16.10)Table 16.1: Approximations of coefficients 𝑣 and 𝑉 in expansios (16.3), (16.16)

𝑛
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

𝑣𝑛
1.0000000000000000−0.5647228383177324
0.3381775868511833−0.2103313021386278
0.1344548790521098−0.0877843886012191
0.0582880930830947−0.0392407117837278
0.0267232860342981−0.0183765205976376
0.0127420898467766−0.0088986329515697
0.0062531995639749−0.0044181328624397
0.0031365295362696−0.0022361213774487
0.0016001999145218−0.0011489818761273
0.0008274921384317−0.0005975832172069

𝑉𝑛
1.0000000000000000
0.5647228383177324
0.2996461813840881
0.1559323904892543
0.0803518797481544
0.0411584960662439
0.0209985209544120
0.0106825803202636
0.0054228810223159
0.0027482526618683
0.0013909151872678
0.0007031815862125
0.0003551700677648
0.0001792537427482
0.0000904088765718
0.0000455725430285
0.0000229602263218
0.0000115627707503
0.0000058201696570
0.0000029289688393

where 𝑥1 ≈ 1.25155147882219 is solution of equation Φ(𝑥1 ) = 1. The
readers are invited to verify, that this tetration satisfies the conditions,formulated in the section 1 of this chapter. For real values of the argument, graphic of this function is shown in figure 16.1. The complex map√of tetration to base 𝑏 = 2 is shown in figure 16.2.
Tetration by (16.10), is periodic; the period 𝑃 is pure imaginary,

𝑃 = 𝑃 (tet√2 ) = −

2𝜋i
2𝜋i
=
−
≈ 17.14314817935485 i (16.11)
ln(ln(2))
ln2 (2)

I remind, the double logarithm ln2 (2) = ln(ln(2)), but does not meanln(2)2 , according to notations declared at the beginning of this Book.
As it is claimed above, tetration tet√2 (𝑧) is holomorphic in the strip|ℜ(𝑧)| ≤ 1. The rangeof holomorphism is much wider than this strip.
√
Tetration to base 2 is holomorphic in the whole complex plane, except

219

8
6
4

𝑢=

3 .8
𝑢=4

2

𝑢=2

𝑣 =1
𝑣 =2

0 cut
−2

0 .2

𝑣=
1.61.8𝑢 =𝑢 =𝑣 =0−4

𝑣 = −1𝑢=2−8

𝑢 = 3.2
𝑢=3

𝑢=4
3 .8
=
−6 𝑢

𝑣 =0

𝑢=
𝑢 = 2 .4
2 .2

𝑢=3

𝑦

𝑣 =0

.4
0
−
=
𝑣
0.2
−
𝑣=

−8 −6 −4 −2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2tetmap.jpg
Fig. 16.2: 𝑢+i𝑣 = tet√2 (𝑥+i𝑦)

the countable set of branch points and the cut lines{𝑧 ∈ C : ℜ(𝑧) ≤ 2, ℑ(𝑧) = 𝑛ℑ(𝑃 ) , 𝑛 ∈ N}

(16.12)

Outside these cuts, tetration approaches the fixed points of the corresponding logarithm, to 2 or to 4, at the increase or decrease of the realpart of the argument, respectively. For any real 𝑦 ,lim tet√2 (𝑥+i𝑦) = 2

𝑥→+∞(16.13)and for 𝑦 ̸= ℑ(𝑇 )𝑛, 𝑛 ∈ N,

lim tet√2 (𝑥+i𝑦) = 4

𝑥→−∞(16.14)For real 𝑥, function tet√2 (𝑥) is defined at 𝑥 > −2. In point −2, thefunction has logarithmic singularity, as tetrations to other values of base.
220

0 .2
=
𝑣
0.4
=
𝑣 0 .6
𝑣= 1
𝑣=

6
4

𝑢 = −2

−3

0

8

𝑣 =0
𝑢=

𝑣=

𝑦

2

𝑢=

−2
−4

cut

𝑣 =0

𝑢=

𝑣 =0

0

−8

−
3

−8 −6 −4 −2

0

2

6

8

𝑢=

4

𝑣=

−6

8
.8
𝑢 = −2𝑢 = −2

−1.

−2
.
𝑢=− 2
2.4
𝑢 = −2.60𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2atemap.jpg
Fig. 16.3: 𝑢+i𝑣 = ate√2 (𝑥+i𝑦)

The function grows from −∞ at −2, passes through points (−1, 0) и
(0, 1), and continue to grow, approaching its limit value 2 at +∞.

√There are several reasons, why tetration to base 𝑏 = 2 is especially√interesting. Namely for this base 𝑏 = 2, both real fixed points of log𝑏
have integer values. In addition, namely for this base, the graphic offunction in figure 16.1 seems to be symmetric with respect to reflection𝑥 ↔ −𝑦 . In order to stress this illusion, the additional line 𝑦 = −𝑥 isdrawn in figure 16.1. For analysis of the illusion mentioned, the inverse
function should be constructed; I mean, arctetration ate√2 = tet√−1. The
2complex map of this arctetration is shown in figure 16.3. This function
is described in the next section.

221

3

√

Arctetration to base 𝑏 = 2

For the inverse function of tetration, I use name arctetration, in analogy
with arcsin, arccos and arcBessel; ate𝑏 = tet−1
𝑏 . Arctetration is abel√function of exponent and satisfies the Abel equation. For base 𝑏 = 2,
this equation can be written as follows:(︁(︁√ )︁𝑧 )︁
𝐺
2= 𝐺(𝑧) + 1(16.15)

One of solutions of equation (16.15) is arctetration 𝐺 = ate√2 . Complex
map of arctetration is shown in figure 16.3.

√

As other abelfunctions, arctetration to base 2 can be evaluated throughits asymptotic expansion, using the Abel equation (16.3) to bring theargument into the range, where the fruncated expansion provides therequired precision. The asymptotic expansion for the arctetration canbe obtained with the asymptotic expansion of tetration. Also, the same
expansion can be obtained directly from the Abel equation (16.3); weshould add the constant to the solution in order to satisfy conditionate√2 (1) = 0.
The two methods mentioned above lead to the same expansion

(︃ 𝑀
)︃
∑︁
1
𝑔(𝑧) = ln
𝑉𝑛 · (𝑧 −2)𝑛𝑘𝑛=1

(16.16)

where 𝑉 are constant coefficients. Constant 𝑘 = ln2 (2) is the same, as
in the expansion (16.3)-(16.5), see equation (16.7). Then, asymptotic
solution 𝐺 of the Abel equation can be written as follows:

𝐺(𝑧) = 𝑔(𝑧) + 𝑂(𝑧 −2)𝑀 +1(16.17)

Substitution of this expansion into the Abel equation (16.15) leads tothe chain of equations for coefficients 𝑉 ; in particular,

𝑉1 =
𝑉2 =
𝑉3 =

(16.18)1ln(2)−𝑣2 = 41 1−ln(2)ln(2)2
1+2 ln(2)24 (1−ln(2))2 (1+ln(2))≈ 0.56472283831773236365(16.19)≈ 0.29964618138408807683(16.20)Approximations of coefficients 𝑉 are collected in the second (and last)column of table 16.1.
With asymptotic expansion (16.16), the solution 𝐺 of the Abel equation(16.15) can be written as limit

𝐺(𝑧) = lim 𝑔
𝑛→∞

(︁

)︁

exp√2𝑛 (𝑧)222−𝑛(16.21)

Here, in the argument of function 𝑔 , function exp√2 is iterated 𝑛 times.
In order to make arctetration ate√2 , not only the asymptotic properties
should be taken into account, but also the value at unity. So, I define
tetration asate√2 (𝑧) = 𝐺(𝑧)−𝐺(1) ≈ 𝐺(𝑧)−1.25155147882219(16.22)that leads to the correct value ate√2 (1) = 0. In previous publication[61], This arctetration is denoted with symbol 𝐹2,1 −1 ; constant 2 in thesubscript indicates the fixed point of the transfer function, at which theregular iteration is constructed, and content 1 in the subscript indicatesits value at zero. As usually, the upper index indicates the number of
iterate.
In figure 16.3, at the complex map of arctetration, its periodicity is seen.
This periodicity follows from the representation of arctetration throughlimit in equation 16.3.
√ The period 𝑃 is determined by the period of the
exponent to base 2;

𝑃 = 𝑃 (ate√2 ) =4𝜋i≈ 18.129440567308775239 iln(2)(16.23)

Imaginary part of this period is slightly greater, than that for tetration
to the same base, see equation (16.11).
In figure 16.3, the isolines are reproduced at the translations along theordinate axis for ℑ(𝑃 ). In addition, due to the real-holomorphism,
the maps of tetration and arctetration are symmetric with respect toreflection from the real axis, id est, with respect to the up side downflip.
In wide range of values of 𝑧 , the identity

tet√2 (ate√2 (𝑧)) = 𝑧(16.24)is valid. This range is shaded in figure 16.4. Technically, the shading isrealised as complex map of the left hand side of equation (16.24), treatedas function of 𝑧 and plotted in coordinates 𝑥 = ℜ(𝑧) and 𝑦 = ℑ(𝑧). Whilethe equation (16.24) holds, the levels of the constant real part and levelsof the imaginary part are parallel to the coordinate axes and form theuniform rectangular grid, that at the poor resolution looks as shading.
However, the relation (16.24) cannot hold in a strip wider than the
period of function ate√2 ; so, the upper part and the lower part of thedomain of the map are not shaded in this way. In addition, the range of
223

𝑦
8
6
4
2
0

cut

𝑣 =0
𝑢=0

−2
−4
−6
−8
−8 −6 −4 −2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2tetatemap.jpg
Fig. 16.4: 𝑢+i𝑣 = tet√2 (ate√2 (𝑥+i𝑦))

validity of equation (16.24) is limited at the right hand side with levels

ℑ

(︁

)︁

exp𝑛√2 (𝑥+i𝑦)

=±

2𝜋
|𝑃 |
=±
2
ln(2)

(16.25)drawn above the map in figure 16.4 for integer 𝑛 = 0, 1, 2, 3, 4. In thementioned right hand side of the figure, relation (16.24) also is not valid.
Figure 16.4 can be considered as verification,
validation, test of imple√
mentation of arctetration to base 2. These properties and the implementations of tetration and arctetration allow to analyse approximate
the symmetry 𝑦 = −𝑥 of graphic 𝑦 = tet√2 (𝑥) shown in figure 16.1.
The apparent symmetry of the plot in figure 16.1 had been declared in
the preamble of this chapter. The consideration
had been postponed√
until tetration and arctetration to base 2 are described. The approxi-

224

mate symmetry mentioned means, that for 𝑥 > −2,

tet√2 (𝑥) ≈ −ate√2 (−𝑥)(16.26)√

Properties of tetration and arctetration to base 2 indicate, that theexact equality in (16.26) can take only at the set of measure zero, due tovery simple and pretty fundamental reason: tetration in the left handside and atctetration in the right hand side of (16.26) have different(incompatible) periods.
Period of function in the left hand side of equation (16.26), see equation(16.11), is 𝑃 ≈ 17.143 i, while period in the right hand side of equation (16.26), see equation (16.23), is 𝑃 ≈ 18.129 i. The different periods
indicates, that these are different functions. If two holomorphic functions coincide at the segment of finite length, they should coincide in
the whole range of holomorphism. Hence, there is no exact equality in(16.26), for the most of 𝑧 , nor for the exact symmetry in figure 16.1.
Deviation from the exact symmetry can be characterised with function(16.27)

devia(𝑥) = tet√2 (𝑥) + ate√2 (−𝑥)It is shown in figure 16.5 with dashed line. Where the symmetry be
exact, the dashed curve should follow the abscissa axis.

𝑦
0.01
−2

−1

𝑦 = tet√2 (𝑥) + ate√2 (−𝑥)
1

2

3

4

5

6

𝑥

7

−0.01
−0.02
−0.03

𝑦 = tet√2 (−ate√2 (𝑥)) + 𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sqrt27u.png
Fig. 16.5: Precision of “symmetry” of figure 16.1: 𝑦 = devia(𝑥) by (16.27), dashed,and 𝑦 = devib(𝑥) by (16.28), solid

Also, the deviation is characterized with function

devib(𝑥) = tet√2 (−tet√2 (𝑥)) + 𝑥
225(16.28)

This dependence is shown in figure 16.5 with solid line. Expression−tet√2 (𝑥) approximate function ate√2 (−𝑥). Again, at the exact symmetry, the solid line would be just abscissa axis. In such a way, figure
16.5 indicates the range of validity of the statement about the symmetry:
it reproduce√ of order of 2 significant figures of tetration or arctetration
to base 2; however, at the segment from −1 to 0 in figure 16.5, the“symmetry” holds with 4 signifivant figures; this is pretty sufficient tocause the illusion of symmetry in figure 16.1.
Similarity of dependences 𝑦 = tet√2 (𝑥) and 𝑦 = −ate√2 (−𝑥) for real 𝑥
may look occasional. However, on the other hand, it is unavoidable forthe following reasons. Every tetration to base 𝑏 > 1 has logarithmic singularity at point −2; the graphics approach vertical line 𝑥 = −2. Graphics of all these tetrations pass through points (−1, 0) and (0, −1), whichcorrespond to the symmetry discussed. In addition, for 𝑏 < exp(1/e),
all the graphics have the horizontal asymptotic for large values of theaugment, they approach some positive quantity (which is fixed pointof logarithm). For some value of base, this quantity is 2, that√ corresponds to the apparent symmetry. This value of base is just 𝑏 = 2, thisbase is chosen as an example in this chapter as illustration behaviour oftetration and arctetration to base 𝑏 at 1 < 𝑏 < exp(1/e).
In years 2009-2010, the apparent symmetry of graphic in figure 16.1
caused hard discussion. The opponents had claimed, that the symmetry
is obvious and does not require any verification. (Before, I had observed
so strong believe in the wrong and absurd statements only in the USSR;
Soviet veterans had insisted on concepts of sovetism, being unable to seeinternal contradictions of it.) To convince the opponents, Henryk andI had elaborated two independent demonstrations, that the exact symmetry cannot take place. Both these proofs are presented in publication[61].
Readers are invited to invent some real-holomorphic function with graphicthat passes through points (−1, 0) and (−0, 1), and exponentially approach to the vertical line 𝑥 = −2 and horizontal line 𝑦 =√2. I suspect,such a function will be pretty similar to tetration to base 2.
With√the suggestion above, I finish the description of arctetration tobase 2. At lest in some vicinity of the half-line 𝑧 < 2, relation (16.24)
is valid, and the pair (tetration,arctetration) can be used to iterate theexponent. These iterates are considered in the next section.

226

4

Again iterate exponent to base√2√

The real-holomorphic Iterates of exponent to base 2 are considered
above, in Chapter 9, for large values of positive part of the argument,with functions SuExp√2,5 and AuExp√2,5 . Those iterates are presented
first (and shown in figure 9.8), because they look similar to iterates
of other growing functions, considered in the first half of this Book.
However, tetration and arctetration, constructedin this chapter, also√
can used to iterate the exponent to base 2. In this section, I show,that these iterates look similar in vicinity of the interval (2, 4); but far
from this interval, the deviation becomes strong.
The “regular iteration” described above, allows to iterate the function,and iterates are regular in vicinity of the fixed point of the transfer
function, used to construct the superfunction and the abelfunction. Bot
these iterates may be not regular (have singularity, branch point) at theother fixed points of the same transfer function. Below, the illustrationof this statement is presented.

√

Iterates of exponent to base 2 constructed with the infinitely growing
superfunction SuExp√2,4 by (9.11),(9.12),(9.13) are shown in figure 9.8.
Similar iterates can be constructed also with tetration, described in thischapter,(︀)︀𝑛exp√2,d(𝑧) = tet√2 𝑛 + ate√2 (𝑧)(16.29)Here, symbol “,d” in the subscript indicates, that the lower, “down”
fixed point of the transfer function is used for the asymptotic of thesuperfunction.
For real values of argument, iterates exp√𝑛2,d by (16.29) are shown infigure 16.6 for various real values of 𝑛. This figure is analogy of figure 9.8,that represents the similar iterates built up with the infinitely growing
superexponent SuExp√2,5 and corresponding abelexponent AuExp√2,5 .
Graphics in figures 16.6 and 9.8 look similar. The thick curves, for
the integer iterates are, indeed, the same. However, for the non-integer𝑛, the iterates also look similar, the curves in figure 16.6 seem to bejust extension, continuation of those in figure 9.8. In the intermediate
range, 2 < 𝑥 < 4, visually, the iterates exp√2,d (𝑥), evaluated throughthe tetration tet√2 and arctetration ate√2 , seems to be the same, as
iterates exp√2,u (𝑥), evaluated through the superexponent SuExp√2 andabelexponent AuExp√2 . Then I saw this coincidence first time, it looked
strange, counter-intuitive and therefore interesting. The matter is, that227𝑛𝑛 == 2
1

𝑦

0
=
𝑛 −1
𝑛=

6
5
4
3

𝑛 → +∞
1 𝑛=2
0 𝑛=1
2

𝑛 = 0.5−2 𝑛 = 0.3
0 .2
=
𝑛
−3
0.1
=
𝑛
−4

−6 −5 −4 −3 −2 −1

𝑛 = −1𝑛 = −2𝑛 → −∞

0𝑛 = −0.4

−6

𝑛=

𝑛=

−5

−0
𝑛 = − .1
0.2

−1

0

1

2

3

4

5

6

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2eitet.jpg
𝑛Fig. 16.6: 𝑦 = exp√(𝑥) for various 𝑛2,utwo different holomorphic functions cannot coincide at the interval of
finite length. If they are identical at the part of the real axis from 2 to 4,
then they must coincide in the whole connected range of holomorphism.
As an example, I consider the case 𝑛 = 1/2; id est, the iterates of
exponent number half. The iterate constructed with growing exponent,
1/2
1/2
id est, exp√2,u had been shown earlier in figure 9.9. The iterate exp√2,d ,constructed with tetration and arctetration, is shown in figure 16.7.
These two maps are not the same. The second of them is periodic (withperiod 4𝜋i/ ln(2) ≈ 18.12944 i), while the first one is not. These twomaps look similar only in vicinity of the interval (2,4) at the real axis.
I felt myself confused about the identical behaviours of the half iterates
along the interval (2, 4) of functions, that have different behaviour in

228

𝑦

cut
cut
cut
cut
cut

𝑣 =0

8
6

−2

.2

𝑣 =0

𝑣

1

𝑢=0

𝑣 =0

𝑢 = −10

𝑢 = −1.2

2

𝑣=

2
= −0.

𝑢=1
𝑢=2

4

cut

−4
−6
−8

𝑣 =0
−10

−8

−6

−4

−2

0

2

4

6

8

cut
cut
cut
cut
cut

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2q2map600.jpg
1/2
Fig. 16.7: 𝑢+i𝑣 = exp√
(𝑥+i𝑦)
2,dthe complex plane. I thought, that I made an error implementing thesefunctions. I even considered the absolutely fantastic hypothesis that
I see the traces of the Mizugadro number 1 , that reveals the internal
contraction in the system of postulates of arithmetics (that is used inmathematical analysis and, in particular, in the theory of holomorphic
functions). I had√ prepared the explicit plot the half iterates of the
exponent to base 2, evaluated through the tetration and that evaluatedthrough the super exponential SuEx√2 ; this plot is shown in figure 16.8,
and looked at the zoom-in of the central part; then at the zoom-in of thatzoom-in, and so on, but I could not see deviation of curve 𝑦 = exp√2,u (𝑥)from curve 𝑦 = exp√2,d (𝑥) .
Searching for the error, as a working hypothesis, I assumed, that the
precision of variables complex double (which is of order of 15 decimal
1 https://mizugadro.mydns.jp/t/index.php/Mizugadro_number

https://budclub.ru/k/kuznecow_d_j/mizugade.shtml Mizugadro’s number (2010-2011)

229

𝑦
6

1/2

𝑦 = exp√2,u (𝑥)

5
4
3
2
1
1/2
𝑦 = exp√2,d (𝑥)−5 −4 −3 −2 −1

00𝑦 = 1024 𝐷(𝑥)
1

2

3

4

5

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2srav.png√
Fig. 16.8: Comparison of the half iterates to base2, constructed at fixed point 2(dashed) and at fixed point 2 (solid curve). In the interval between these two points,24
the difference (16.30) is shown, scaled with factor 10 .

1/2

digits) is not sufficient to see the difference between exp√2,𝑑 by formula
1/2(16.29) and exp√2,𝑢 by formula (9.24). I begun to compute the iteratewith a hundred decimal digits; the exact formulas and abilities of Maple
and Mathematica allow this. The deviation happened to be in 25th
digit. This deviation can be characterised with difference
1/2

1/2

𝐷(𝑥) = exp√2,u (𝑥) − exp√2,d (𝑥)(16.30)This difference is shown at the bottom of figure 16.8. In order to see
it, I had to scale it with factor 1024 , as it is marked in the figure. For|𝑧−3| < 2, function 𝐷(𝑧) can be approximated with the 7 parameter fit

(︁
)︁
−25
2
˜
𝐷(𝑧) = 2.48·10 (𝑧 −2)(4−𝑧) 1 + 0.120(𝑧 −3) + 0.006(𝑧 −3) ×(︁)︁
2sin .747−.068(𝑧 −3)+0.007(𝑧 −3) +𝑝4 ln(4−𝑧)+𝑝2 ln(𝑧 −2) (16.31)
where 𝑝4 = 2𝜋/ln(2 ln(2)) ≈ 19.23614904204285
and 𝑝2 = −2𝜋/ ln2 (2) ≈ 17.14314817935485
correspond to periods of the two superfunctions of exp√2 , built up on the
fixed points 4 and 3, see equations (9.6), (9.7) (16.11). Тhis fit provides
of order of two significant figures; at figure 16.8, the curves for 𝐷 and˜ almost coincide.
for 𝐷
230

After to plot the bottom curve in figure (16.8), I realised, that it is firstcase in my life, when the double precision, id est, 15 significant digits,
happened to be not sufficient to see the difference between two functions,which have no small parameters. This example gives a good lesson: thenumeral calculus with high precision may provide serious evidence torefute, reject (or to verify) a conjecture. Whenever the rigorous proof
is available of not, the numerical testing should be applied. The TORI
axioms provide a good hint for the revision.
It should be noted, that my attempts to use the Maple-10 software forvisualisation of difference 𝐷 by (16.30) failed. I could not find way toplot graphics with precision better than just “float” 2 .
In order to plot the beautiful figure (16.30), I had to save values of function 𝐷 as a table, and then export this table to the C++ program. I
hope, now there exist more straightforward ways to do the same; in particular, use of the fit 16.31 provides the “quick and dirty” realisation. I
expect, using the precise numerical solution, fit 16.31 can be significantly
improved; the readers are invited to do this as an exercise.
Iterate of a function, regular at some of fixed points, often is singularin another fixed point. Some exceptions, when the superfunction isexpressed with elementary function, are mentioned in Chapter 4.
In such a way, for a given transfer function, there may exist many superfunctions, and some of them may be arguably declared as a “true”
or “principal” superfunction. For the case of transfer function 𝑇 =exp√2 , the four real-holomorphic superfunctions with various exponential asymptotics are considered in the next section.

5

Four superexponents to base√2

On the base of consideration of previous chapters, one can built-up thefour different real-holomorphic superfunctions for the transfer function
2 The poor precision of the graphic procedures in Maple-10 is described in the poem https://en.wikisource.org/wiki/Maple_and_Tea Maple and tea. This is one few my texts, that are
not yet removed from wikisource with pretext of protection of my author rights. This is common
practice at wikisource and other sites of Wikimedia projects: the Soviet veterans promote sovetism
and remove texts of anti-Soviet authors with any absurd pretext; often, the claim for the violationof the author rights is used, even in cases, when the author gave permission to publish the files
providing the free ("copyleft") licence. The permission to use the author’s file is usually removed
together the file. While the soviet veterans act as trolls and vandals in wiki-projects, I loaded the
copy also to https://budclub.ru/k/kuznecow_d_j/maple.shtml

231

𝑦
8
6

𝑦 = 𝐹4,5 (𝑥) = SuExp√2,5 (𝑥)4𝑦 = 𝐹2,3 (𝑥), 𝑦 = 𝐹4,3 (𝑥)
2
−8

−6

−4

−2

𝑦 = 𝐹2,1 (𝑥) = tet√2 (𝑥)𝑦 = 1024 𝑑42 (𝑥)
2
4
6
8

𝑥

−2
https://mizugadro.mydns.jp/t/index.php/File:Sqrt2sufuplot.png
Fig. 16.9:

Four superexponents to base√2

𝑇 = exp√2 . These superexponentials are shown in figure 16.9 and discussed below.
Here I compare four functions; in publication [61], they are are called
𝐹2,1 , 𝐹2,3 , 𝐹4,3 and 𝐹4,2 . Each of them is real-holomrphic solution of thetransfer equation(︁√ )︁𝐹 (𝑧)𝐹 (𝑧 +1) =2(16.32)

The first number in the superscript indicates the limiting value, that thefunction approaches exponentially; it is any of the two fixed points ofthe exp√2 , id est, either 2 or 4. The second number in the superscript
indicates value of this function at zero.

√

Function 𝐹2,1 = tet√2 is tetration to base 2; the curve for 𝐹2,1 is borrowed from the figure 16.1. Properties of this function are considered√
above in this chanter. It is superfunction of exponent to base 2, built
up with the regular iteration at fixed point 2. As tetration to any otherbase, it takes value unity at zero.
Function 𝐹4,5 = SuExp√2,5 refers to formula (9.13). This function, to−1
gether with its inverse function 𝐹4,5
= AuExp√2,5 is used to built-up√
iterates of exponent to base 2; and these iterates grow up infinitely232along the real axis. Fixed point 4 is used as the asymptotic value atminus infinity. With the appropriate translation along the real axis, thecondition 𝐹4,5 (0) = 5 is achieved.
Function 𝐹2.3 is tetration with displaced argument,

𝐹2,3 (𝑧) = tet√2 (𝑧 +𝑧2,3 )(16.33)𝑧2,3 = ate√2 (3+i𝑜) ≈ −3.3834692659172254 + 8.5715740896774228 i
However, 𝐹2,3 has the same periodicity, as tetration 𝐹2,1 .
Function 𝐹2,3 is growing superexponent with displaced argument

𝐹4,3 (𝑧) = SuExp√2,5 (𝑧 +𝑧4,3 )(16.34)𝑧4,3 = AuExp√2,5 (3+i𝑜) ≈ 3.015784890490347+9.618074521021425 i
Along the real axis, functions 𝐹2,3 and 𝐹4,3 decrease from 4 at minus
infinity to 2 at plus infinity. In figure 16.9, curve 𝑦 = 𝐹2,3 overlaps wellwith curve 𝑦 = 𝐹4,3 . The deviation is smaller, than the thickness oflines, and it is small compared to size of atoms, of which this book (orthe screen where it is watched) is built. In otter to show the deviation,
denote it with𝑑42 (𝑧) = 𝐹4,3 (𝑧) − 𝐹2,3 (𝑧)(16.35)Tn figure 16.9, the thin line shows 𝑦 = 1024 𝑑42 (𝑥) ; I scale values of thisdifference for 24 orders of magnitude, to make it visible. This similarity
takes place only in vicinity of the real axis. The functions have different
periods, and one of them has singularities; so, they must be pretty
different somewhere.
Similarity of functions 𝐹4,3 and 𝐹2,3 determines
the similarities of corre√
sponding iterates of exponential to base 2. These iterates are shownin figures 9.8, 9.9, 16.6 and 16.7. For real values of the argument, thehalf iterates are compared also in figure 16.8. I expect, for applicationin physics (where the precision usually does not exceed 20 decimal digits), any of the two iterates is declared as the “true iterate”. However,for some applications (for example, if the model refers to the complex
numbers), the difference may be important, and the fixed point shouldbe specified.

233

6

𝑢=

2

𝑣 =2

1

𝑢=

4

4

𝑣 =3

𝑢=

8

3

𝑢=2

𝑦

𝑣 =1
𝑣 =0
𝑢 = 0 𝑣 = −1

0
−2

cut

−4
−6
−8

−10

−8

−6

−4

−2

0

2

4

6

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2diimap80.jpg
Fig. 16.10:

6

8

𝑥

i
𝑢+i𝑣 = exp√2,d(𝑥+i𝑦) by (16.36)

Complex iterates

When the draft of the Russian version of this Book had been completed,
I found, that the book had no maps of the complex iterates. I had
declared that I can calculate any real or even complex iterate, but all
the examples refer to real iterate. I fill this gap in this section. Here I
describe two iterates number i. As this i appear with Roman font; one
may guess, that it is not variable, but a constant, square root of −1.
Figures 16.10 and 16.11 show the complex maps of iteratesi
exp√2,d(𝑧) = tet√2(︁)︁

(16.36))︁
2,5 (𝑧)(16.37)i + ate√2 (𝑧)andi(𝑧) = SuExp√2,5
exp√2,u(︁i + AuExp√

234

𝑣 =3

6

𝑢=

4

1

−1

𝑣 = 100

𝑣=

−6

0
10
− 100
𝑢 =𝑢 = −100𝑣= 𝑣 =0

−4

2 𝑢=5
𝑣=
6
=
1
𝑣= 𝑢
0 𝑢=7
𝑢=8
𝑢=9
𝑢 = 10𝑢=1
𝑢= 2
𝑢 = 14
16

cut

−2

−6

−4

−2

0

2

4

18
𝑢 = 20
𝑢=

−8

−8

4

𝑣=

𝑢=

2
0

𝑢=

8

3

𝑢=2

𝑦

6

https://mizugadro.mydns.jp/t/index.php/File:Sqrt2uiimap80.jpg

8𝑥iFig. 16.11: 𝑢+i𝑣 = exp√
(𝑥+i𝑦) по формуле (16.37)2,uiFunction exp√2,dby (16.36) is built up from the tetration and arctetra-tion to base√i
2, considered in this chapter. Function exp√2,uby (16.37)√
is built up from the growing super exponent to base 2 and the corresponding abelexponent; these functions are considered in chapter 9.
These pairs of functions look similar in vicinity of the interval (2, 4), butthey are pretty different being evaluated far from this interval. As one
could expect, the i th iterates, shown in figures 16.10 and 16.11, are alsosimilar in vicinity of the interval mentioned, but far from this interval,
they deviate strongly.

As in the case of real iterates, each of considered here complex iterates
can be arguably qualified as “true”. In this sense, the are “equal”. In
the similar sense, "all animals are equal" in the novel “Animal’s Farm”

235

by George Orwell 3 In the movel, soon it happens that “All animals areequal, but some of them are more equal than others”. In the similar
sense, one of iterates, either 16.10 or 16.11, may be “more equal”, if any
additional criterion arises from some physical reasons, specifying, forexample, the asymptotic behaviour of the iterate at infinity. If only oneof iterates reproduces the required behaviour, this iterate immediately
becomes “more equal” than another.
The example of this section once again confirms the general observation
about non-integer iterates of a function, that have more than one fixed
point: the regular iterate, holomorphic in vicinity of some fixed point,
has no meed to be holomorphic at another fixed point. The choice tothe fixed point (and choice of the iterate) should involve the additional
requirement, that may arise from the applications. This general rulehold also for complex values of the number of iterate.
I expect, the integer iterates may appear more often that real; and thereal iterate may appear more often than complex. On the other hand,
the mathematical formalism should cover an area, which is wider, thanthat required for the today’s applications. For this reason I consider
the case with complex number of iterate as an important example, thatshows the abilities of the formalism of superfunctions.
Following the lessons I remember since the Soviet school, I wanted to saythat “the formalism of superfunctions is omni-potnent, because it is true”
4. On the other hand, the First TORI axiom prohibits consideration ofomnipotent and almighty concepts in a scientific analysis; such doctrinesand concepts are qualified as religious [69].
Various iterates are available for a transfer function with several fixed
points. As soon, as the non-integer iterate of a holomorphic function
with several fixed point is required, the additional conditions should be
added to the formalism in order to decide, which of the iterates is “moreequal than others”.

3 https://msxnet.org/orwell/print/animal_farm.pdf George Orwell. Animal Farm. 1945.

.. “All animals are equal, but some of them are more equal than others” ..

4 https://www.marxists.org/archive/lenin/works/1913/mar/x01.htm

V.I.Lenin.

The

Three Sources and Three Component Parts of Marxism. Lenins Collected Works, Progress Publishers, 1977, Moscow, Vol.19, p.21-28. .. The Marxist doctrine is omnipotent because it is true.
..

236

7

Not all is done

In this Book, I describe methods, that can be used to build-up (and evaluate) superfunctions, and the iterates that can be expressed through thesuperfunctions and the abelfunctions. The properties of superfunctons
and those of the iterates appear as illustrations of the methods. I expect the Reader to use these methods not only for the transfer functions,considered in this book. In particular, this chapter indicates the interesting (from my point of view) branch for the future research, namely,comparison of superfunctions and the corresponding iterates, that are
regular in various fixed points, and look for some general criteria: inwhich cases, the iterates, built up at different real fixed points of somereal-holomorphic transfer function, behave in the similar way in the interval between these fixed points.
√ They happened to be very similarin the case of exponent to base 2. How about other growing transfer
functions with two real fixed points?
The Readers are invited to repeat the calculus, described in this Book
(and in this Chapter) for other transfer functions. As an example, I
would suggest to experiment with polynomial transfer function.
One example of the polynomial transfer function is shown in raw 5 oftable 3.1, 𝑇 (𝑧) = 𝑧 𝑏 . The readers may confirm, that the primary approximation with the regular iteration at the fixed point 𝑧 = 1 gives the seriesof expansion of the superfunction, that is just expansion of the “outer”
exponent in superfunction exp(𝑏𝑧 ). Up to my knowledge, this is the only
case, where the primary series by the regular iteration converges. The
appropriate choice of the 0th approximation should cut the series at thefirst term. I expect, many of transfer functions can be treated in sucha way, and the corresponding superfunctions and abelfunctions can bebuilt-up. The goal of this Book not to describe all the examples, but toteach the Readers to to it by themselves. So, it is rather collection oftools, than collection of specific properties of the specific functions.

√

Consideration of tetration to base 𝑏 = 2, presented in this chapter,can be generalised to other values of base; in particular, for real values
1 < 𝑏 < exp(1/e). This generalisation is described in the next chapter.

237

Chapter 17
Tetration to base 𝑏 > 1
The previous chapters describe various methods of construction of superfunctions for the transfer functions that have real or complex fixed
points. In particular, the examples of the exponential transfer function√
are considered, 𝑧 ↦→ 𝑏𝑧 for the cases 𝑏 = 2 ≈ 1.44, 𝑏 = exp(1/e) ≈ 1.46
and 𝑏 = e ≈ 2.71 .
In this chapter, I combine methods, described in the previous chapters,
and describe tetration to the real base 𝑏 > 1. For the real values of base,the interpretation of the superfunction, abelfunction and iterates of the
exponent is especially explicit.

1

Approximation of tetration near zero

For the base 𝑏 from the interval 1 < 𝑏 < exp(1/e), tetration tet𝑏 can beevaluated with regular iteration at the lowest (smallest) positive fixed
point of the exponent to this base. For 𝑏 = exp(1/e), it can be evaluatedwith the exotic iteration by equation 10.44. For 𝑏 > 1/ exp(1/e), therepresentation through the Cauchy integral can be used. In such a way,
all the domain 𝑏 > 1 is covered with the efficient algorithms for evaluationof tet𝑏 .
For real values of the argument, graphic 𝑦 = tet𝑏 (𝑥) is shown in figure17.1 versus 𝑥 for various values of base 𝑏 > 1. Similar plot for arctetrationis shown in figure 17.2.
Figure 17.1 for tetration and figure 17.2 for arctetration are generated,using the special approximation of tetration for 𝑏 < 3, by function fit1,
defined with𝑑 = ln(𝑏)238𝑏 = 1 .5

𝑏 = 1.6

𝑏 = 1.7

𝑏 = 1.9𝑏 = 1.8

𝑏=2

𝑏=e

𝑏→∞

8

𝑏 = 10𝑦64e𝑏 = exp(1√/e)

2
0
0

𝑏= 2

𝑏 = 1.4
𝑏 = 1.2
𝑏→1

2

4

6

8

10 𝑥

https://mizugadro.mydns.jp/t/index.php/File:Tetreal10bx10d.png
https://en.citizendium.org/wiki/File:Tetreal10bx10d.png
Fig. 17.1:

𝑦 = tet𝑏 (𝑥) for various 𝑏√𝑞= 𝑑𝑐0 = −1.0018 + 0.1512848482(1.+33.0471529885𝑞−3.51771875598𝑑)𝑞+1+3.2255053261256337𝑞

ln(2)− 12𝑑𝑐1 = 1.1 − 2.608785958462561(1 − 0.6663562294911147𝑞)𝑞 −𝑐2 = −0.96 + 3.0912038297987596 (1+0.60213980487853𝑑)𝑞+
1+4.24046755648𝑑

ln(2)− 23𝑑𝑐3 = 1.2−10.44604984418533 (1+0.213756892843𝑞+0.369327525447𝑑)𝑞−
1+4.9571563666𝑞+7.702332166𝑑
fit1𝑏 (𝑧) = (1+𝑐0 𝑧 +𝑐1 𝑧 2 +𝑐2 𝑧 3 +𝑐3 𝑧 4 )(𝑧 +1) + ln(𝑧 +2) −239

ln(2)− 58𝑑ln(2)− 131
192
𝑑

ln(2)
(1+𝑧)
𝑑
(17.1)

𝑏 = exp(1/e)

𝑏 = 1.5

𝑏= 2

𝑏→1
𝑏 = 1.2
𝑏 =√
1.4

𝑦
10
8
6

𝑏 = 1.6

4

𝑏 = 1.7
𝑏 = 1.8
𝑏 = 1.9

2

𝑏=e

𝑏=2

𝑏 = 10

0
0

2 e

4

6

𝑏→∞

8

10 𝑥

https://mizugadro.mydns.jp/t/index.php/File:Ater01.png
Fig. 17.2: 𝑦 = ate𝑏 (𝑥) for various 𝑏

In order to get approximation (17.1), I expanded the expression

tet𝑏 (𝑧) − ln(𝑧 +2) + ln(2)/ ln(𝑏)(17.2)
1+𝑧
into the Taylor series with powers of 𝑧 for various values of (new) parameter 𝑑 = ln(𝑏). Coefficients of this expansion are approximated asfunctions of parameter 𝑑. Then, tetration is expressed through this expansion. The series is truncated; only few terms are taken into account.
For |𝑧| ≤ 1/2, the resulting approximation provides few correct decimal
digits of tet𝑏 (𝑧). This approximation is used for |ℜ(𝑧)| ≤ 1/2; for othervalues of 𝑧 , value of tetration is represented through its values at theappropriate argument, using either
tet𝑏 (𝑧) = 𝑏tet𝑏 (𝑧−1)(17.3)(︀)︀
tet𝑏 (𝑧) = log𝑏 tet𝑏 (𝑧 +1)(17.4)

or240

dependently on signum of ℜ(𝑧). In such a way, tetration can be evaluated with few decimal digits for moderate values of |ℑ(𝑧)| and |ℑ(𝑏)|.
Yet, I have no similar approximation for larger values of |ℑ(𝑏)|, although,the Cauchy integral can be generalised for various, even complex valuesof 𝑏; the example is considered below in chapter 18. The range of validityof approximation fit1b is wide; In particular, at 𝑏 < 5, the approximation
provides of order of four significant figures. This is sufficient precision
for plotting of the camera-ready pictures, the defects of this approximation are not seen in figure 17.1. However, at 𝑏 = 10, the approximation isa little bit worse; for this value, the primary representation through the
Cauchy integral is used. This refers to only single curve in the picture.
Figure 17.1 shows behaviour of tetration of real argument at various
values of base 𝑏 > 1:
In the interval −2 ≤ 𝑥 ≤ −1, tetration 𝑦 = tet𝑏 (𝑥) has negative values
and grows while 𝑏 increases.
In the interval −1 ≤ 𝑥 ≤ 0, tetration 𝑦 = tet𝑏 (𝑥) has positive value andgrows with increase of 𝑏.
At 𝑥 > 0, tetration 𝑦 = tet𝑏 (𝑥) has positive values and grows with increaseof 𝑏.
In the limiting case 𝑏 → 1, the curve 𝑦 = tet𝑏 (𝑥) approaches the asymptotics 𝑥 = −1 and 𝑦 = 1.
In the limiting case 𝑏 → ∞, curve 𝑦 = tet𝑏 (𝑥) approaches asymptotics
𝑥 = −2 and 𝑥 = 0, and also −2 < 𝑥 < 0 at the abscissa axis.
At all 𝑏 > 1, tetration tet𝑏 (𝑥) is monotonic function, and curve 𝑦 = tet𝑏 (𝑥)passes though points (−1, 0), (0, 1) and (1, 𝑏).
At 𝑏 = exp(1/e), the line 𝑦 = filog(1/e) = e becomes asymptotics. thisline is added to the rectangular grid at integer values of abscissas andordinates.
At 1 < 𝑏 ≤ exp(1/e), with grow of(︀ 𝑥 the)︀curve 𝑦 = tet𝑏 (𝑥) approaches thehorizontal asymptotics 𝑦 = filog ln(𝑏) .
Function filog expresses fixed points of logarithm as function of logarithm of its base. This function is considered in the next chapter. It isessential for evaluation of tetration of complex base. While, suggest tocompare the results of the previous chapters for the real base. This ismatter of the next section.

241

2

Various bases of exponent and the iterates

In this section, the iterates of exponent are compared. These iterates
are considered in the previous sections. Below, I overview the results.

√

Iterates of exponent to bases 𝑏 = e, 𝑏 = exp(1/e) and 𝑏 = 2 are shown infigure 17.3 as functions of the real argument for various values of number
𝑛 of iterate. The curves are drawn for 𝑛 = −2, −1, −0.9, −0.5, −0.1,
0, 0.1, 0.5, 0.9, 1, 2.
The upper picture in figure 17.3 represents the case 𝑏 = e; iterates of thenatural exponents are shown. These iterates are calculated through thenatural tetration tet and natural arctetration ate, I repeat this formula
once again:)︁𝑦 = exp (𝑥) = tet 𝑛 + ate(𝑥)𝑛(︁(17.5)

Similar pictures can be plotted also for other values of base 𝑏 > exp(1/e).
For these values of base, the real iterates are real-holomorphic functionsat least in some vicinity of the real axis. In this area, the iterates of the
exponent are so smooth, as the exponent itself.
As the base 𝑏, the width of the strip along the real axis (where theiterates are holomorphic) decrease; at base 𝑏 = exp(1/e) all the curves
for various iterates pass through the fixed point e. This case is shownin the central picture of figure 17.3. Then, the non-integer iterates of
exponents at argument, larger than e, are not anymore holomorphic
extensions of those for argument, smaller than e. In order to stress this,the curves, plotted through tetration and arctetration, are shown withdashed lines𝑛𝑦 = exp𝑏,d(𝑥) = tet𝑏

(︁)︁𝑛 + ate𝑏 (𝑥)(17.6)

while the solid lines correspond to the iterates, expressed through thegrowing superexpponent SuExp,𝑛𝑦 = exp𝑏,u(𝑥) = SuExp𝑏,3(︁)︁
𝑛 + AuExp𝑏,3 (𝑥)(17.7)

Tetration tet𝑏 to base 𝑏 = 𝜂 = exp(1/e) is determined with (10.44), andthe arctetration to this base can be evaluated through (10.49). In thesimilar way, the growing superexpenent is determined by (10.45) with
SuExp𝑏,5 = 𝐹3 , and the corresponding abelexponent can be evaluatedwith (10.50) at AuExp𝑏,3 = 𝐺3 .
For the real base 𝑏, at 1 < 𝑏 < exp(1/e), The exponent has two real fixed
points (See figures 9.1, 10.1). Each to these fixed points can be used for
242

4

2

4

0.5

𝑛 = −0.9
𝑛 = −1

6

8

𝑥

𝑛=

−2

−1
𝑛=

𝑥

𝑛=−

)
𝑦 = ln(ln(𝑥) 𝑛 = −2

0
𝑦=

.1
0
−

√ n(𝑥)l𝑦=𝑦 = ln ( 𝑥 )

𝑛=2

𝑛=2
𝑛=
1

𝑦

1/e

𝑏=e

0
𝑛=

2
𝑛=1
𝑛 = 0.9𝑛 = 0.5

𝑛=

0 .1
𝑛=

𝑏=e

𝑦 = exp( 𝑥
e )
𝑦 =e
xp(𝑥
)
𝑦= √
exp
(𝑥)

𝑦

n(𝑥)
l
e
𝑦 = n(𝑥))(e l𝑦 = e ln41𝑛=−𝑛 = −2e2p(1 e𝑥/e )64𝑦𝑛 = −2
2

𝑦

𝑛=2

4

8

𝑥

1
𝑛=−
𝑛 = −2)ln 𝑏(𝑥)= ln 𝑏(

𝑛=

√
𝑏= 2

2 e

𝑛 = −2

𝑛=

−0
.5
𝑛=−
1

𝑦 = exe) e
𝑛=2
/
exp(𝑥
=
𝑦
𝑛=1
0
𝑛 = 0.9.5𝑛 = 0 .10𝑛=2𝑛 = −2

𝑛=

−0
.5
𝑛=−
1

𝑛=2
𝑏𝑥
=
𝑦
𝑛=1
0
𝑛 = 0.9.5𝑛 = 0 .1
0
=
𝑛

2

4

6

8

https://mizugadro.mydns.jp/t/index.php/File:E1e14z600.jpg √Fig. 17.3:

𝑦 = exp𝑛𝑏 (𝑥) for various 𝑛 at 𝑏 = e, 𝑏 = exp(1/e), 𝑏 = 2
243𝑥the regular iteration. However, the non-integer iterate, regular at one
fixed point, is singular
at another one. At the bottom picture of figure√
17.3, value 𝑏 = 2 is chosen.
Dashed lines corresponds to𝑛𝑦 = exp𝑏,d(𝑥) = tet𝑏

(︁)︁𝑛 + ate𝑏 (𝑥)(17.8))︁𝑛 + AuExp𝑏,5 (𝑥)(17.9)

while the solid curves refer to𝑛𝑦 = exp𝑏,u(𝑥) = SuExp𝑏,5(︁

For non-integer number 𝑛 of iterate, the fixed point 2 or 4 limits therange of holomorphism of each iterate. In the interval from 2 to 4, eachof the two iterates is holomorphic, and the difference between these
iterate is very small, of order of 10−24 , see figure 16.8. Due to thissmallness, the dashed lines at the bottom picture of figure 17.3 seem tocoincide with the corresponding solid lines.
Similar illusion takes place for the central part of figure 17.3; the dashedlines seem to be continuations of the corresponding solid lines. The
example with exponent show, that, in order to specify the non-integer
iterate, one should choose, establish the asymptotic behaviour of thesuperfunction in the complex plane. Over-vice, there may exist various solutions, and each of them arguably can be declared as the “true”
one. In order to specify the superfunctions, they are considered for thecomplex argument.

3

Dependence of tetration on its base

Graphics of tetration of real argument, shown in figure (17.1), allow toguess, that the dependence of tetration on the base (at fixed argument)
is continuous (and, perhaps, even holomorphic) function. In order toshow, that this refers not only to the real values of the argument, figure17.4 shows the complex maps of tetration
for 𝑏 = 1.5 at left, for 𝑏 =√exp(1/e) ≈ 1.44 at center, and for 𝑏 = 2 ≈ 1.41 at right.
All the 3 maps in figure 17.4 look similar, although different algorithms
are used for evaluation of tetration. In principle, tetration to base 𝑏 =exp(1/e), for moderate values of the imaginary part of the argument,
could be evaluated also through tetration with a little bit smaller or alittle bit larger values of the base 𝑏, as limit 𝑏 → exp(1/e), using thecorresponding representation through the integral Cauchy (for bigger
values) or with regular iteration (for smaller values). The Readers areinvited to calculate tetration to base 𝑏 ≈ exp(1/e) and estimate, how
244

𝑣=

−4
−4 −3 −2 −1

−1

−3

0

1

2

3

𝑥

𝑣 =1
𝑣 = 0.8
𝑣 =0
.6
𝑣=
0 .4
−1
𝑣=
−3 −2 −1

2
0.

𝑣 =0

1 .6
𝑢=
8
𝑢 = 1.
𝑢=2
0 1 2

.2
−0

−2

𝑣 =0
𝑣
4
1. .6 = −0=
𝑢 = 1 .8 .2
𝑢 =1
𝑢
𝑢=2
−3 −2 −1 0 1 2 3 𝑥

𝑣=

𝑣=

−1

𝑣

.2
=0

𝑢
𝑢= =1
1.
4

0

𝑢 𝑢=
𝑢 = 1
𝑢 = = 1 1 .4
𝑢 = 1 . .6
8
2

1

𝑣 =1
𝑣 = 0 .8𝑣=
𝑣 = 0.6
0.
4

2

0.8
=
𝑣
0 .6
𝑣=
4
𝑣 = 0.
𝑣 = 0.2𝑣 =0𝑣 = −0.2
𝑣 = −0.4

13𝑣 = −1 𝑢 =

𝑣 =1

𝑦

3

𝑥

https://mizugadro.mydns.jp/t/index.php/File:E1efig09abc1a150.png√
Fig. 17.4: 𝑢+i𝑣 = tet𝑏 (𝑥+i𝑦) for 𝑏 = 1.5, 𝑏 = exp(1/e) and 𝑏 = 2

many significant figures can one achieved in such a way, assuming, thatthe arithmetics with finite precision is used.
At 𝑏 → exp(1/e), the efficiency of evaluation of tetration (both throughthe Cauchy integral and through the regular iteration) reduces. For thisreason, Henryk Trappmann had expected, that tet𝑏 is not continuousfunction of 𝑏 at point 𝑏 = 𝜂 = exp(1/e). For this reason, Henryk wanted
the asymptotic expansion namely for 𝜂 = exp(1/e), and it had been
done 1 . I had suggested the expansion (10.36) and plotted pics for𝑏 = exp(1/e), and Henryk had arranged a lot of mathematical deduction
around it [80].
I hope, the Reader already understands, how to guess the expansion for
the exotic iterates, and can write the similar expansion of superfunction for any other transfer function, as soon as such an superfunction
will be requested for any application. The inversion of the series gives
the expansions (and the precise approximations) for the corresponding
abelfunctions.
I hope, with the tools above, the colleague can evaluate any iterate ofany holomorphic transfer function, not only real, but also complex. The
examples with iterates number i are shown in figures 16.10 and 16.11
for the transfer function 𝑇 = exp√2 .
After publications of the results presented above, the colleagues at the
Henryk’s forum had agreed, that the complex iterates, and in particular,
those of the exponent to various real base 𝑏 > 1, can be evaluated in a
pretty regular way. However, There were some doubts about iterates ofthe exponent to the complex base. This case is considered in the next
chapter.
√

1 That happened soon after the article about four real-holomorphic superexponentials to base

2 had been submitted to Mathematics of computation [61]; as usually, the appetite comes whileeating.
245

Chapter 18
Tetration to complex baseHere, tetration to complex base is considered. Id est, superfunction for
the transfer function 𝑇 (𝑧) = 𝑏𝑧 , where base 𝑏 is not real. In order tohandle this case, I need an additional special function "filog"; it is also
described in this chapter.
In principle, the superfunction of the exponent to complex base can beconstructed with regular iteration, in the similar way, as the tetration to√base 2 is constructed in Chapter 16. However, the important question
is, at which of the fixed points, the superfunction should be regular, andwhich of possible superfunction should be qualified as tetration.
In addition, at some values of the base, the exponential asymptotic
solution has real part of the increment zero or close to zero; this makes
the application of regular iteration difficult, if at al. One of such cases is
considered below in more details, as an example. The representation oftetration through the Cauchi integral is not sensitive to the real part of
the asymptotic increment of the solution; so, such a representation gives
the efficient way of evaluation of tetration to complex base. However,
the asymptotic behaviour of tetration in the upper and in the lower
complex plane should be specified.
The main idea of this chapter is to make superfunction that approaches
one of the fixed points of the exponent at the upper half of the complex
plane, and to another fixed point at the lower part of the complex halfplane, using the assumption, that, for the complex base, the imaginary
part of the asymptotic increment is not zero.
Question about the fixed points is important (as in the case of any
other superfunction), and it should be considered. This consideration ispresented in the following section.

246

𝑦

8

𝑢=

0

6

0 .4
𝑢=

2

𝑢=

0 .2

4

0

𝑢=

𝑢=0

2

−2

𝑣=

−0.
𝑢=

−4

−6

−0
.2
𝑣=

−4

−2

0

−6

𝑣=

−8

0

2

4

6

8

https://mizugadro.mydns.jp/t/index.php/File:Filogbigmap100.png
Fig. 18.1:

1

0 .2

𝑣 = −0.4

−8

−10

−0
.2

𝑥

𝑢+i𝑣 = filog(𝑥+i𝑦) , as solution 𝐿 = filog(𝐵) of (18.1)

Fixed points of logarithm

This section is dedicated to relation between the base 𝑏 of the exponential and its fixed point 𝐿. Let 𝐿 = filog(𝐵) be solution of equationln(𝐿) = 𝐵𝐿

(18.1)

Complex map of function filog is shown in figure 18.1. The zoom-in of
the central part of this map is shown in figure 18.2. Let

𝐵 = ln(𝑏)(18.2)

Then filog(𝐵) determines the fixed point 𝐿1 of logarithm to base 𝑏;
another fixed point 𝐿2 is determined with the complex conjugation:

𝐿1 = fllog(𝐵)𝐿2 = filog(𝐵 * )*247

(18.3)(18.4)

1

0 .2

𝑣=

.8

𝑢 = 0.6

𝑢=

𝑦

𝑢=

𝑢=0

0

0 .8

0

−1
−1

1/e

0

1

𝑣 =1

𝑣=

2

𝑣 = 1.2

𝑢 = −0.2

𝜋/2

𝑥

𝑦

4
10

3
10

𝑣=
0.4

2

0 .2

𝑢 = 1.

110

𝑢=1

𝑢 = 0 .8

𝑣=

𝑢=1
.4
𝑢=
1 .6

2
10

0

𝑣 =1
𝑣
𝑣 == 2
5 3
=
𝑢 =4
𝑢 3
𝑢=
𝑢=2
𝑢=1
𝑢=
0

−1
10

−1
5

−3
10

−0.4

𝑣 =1

−1
2

−0.3

−0.2

𝑣=

𝑣=

−1
𝑣 =0

2
−

−2
5

−0.1

0

0.1

0.2

0.3 1/e 0.4

https://mizugadro.mydns.jp/t/index.php/File:Figlogzo2t.jpg
Fig. 18.2: 𝑢+i𝑣 = filog(𝑥+i𝑦), zoom-in from the central part of figure 18.1

248

𝑥

Function filog can be expressed through the special function Tania by(5.3) as follows:(︀)︀Tania ln(𝑧) − 1 − 𝜋ifilog(𝑧) =(18.5)−𝑧Note, that here, namely Tania is used, but not WrightOmega, whichlooks similar to Tania in vicinity of the real axis. The Readers areinvited to try to reproduce figure 18.1, using WrightOmega instead of
Tania, and look, what does it give instead of the beautiful map.
Function filog determines the fixed points of logarithm (which are also
fixed point of the exponent) to base 𝑏:(︀)︀
𝐿1 = filog ln(𝑏)(︀)︀*𝐿2 = filog ln(𝑏* )(18.6)(18.7)

At ±i∞, tetration should approach these fixed points. This assumption
to use the Cauchy integral for the definition (and evaluation)

2

Tetration to the Sheldon baseThis section describes the tetration to the Sheldon base,𝑏 = 1.52598338517 + 0.0178411853321 i(18.8)

This number is named after Sheldon Levenstein. In 2015, Sheldon had
expected, that the namely this base causes difficulties at evaluation oftetration. It was the only request from colleagues to evaluate tetration
to the specific complex base; and this request had been fulfilled.
Complex map of tetration to the Sheldon base is shown in the top pictureof figure 18.3. Explicit plot of this function is shown in the central
picture of that figure. The bottom picture shows the explicit plot of
tetation to Sheldon base along the imaginary part. This tetration has
complex values; so, the graphics are drawn for the real and for theimaginary parts.
In this section I assume, that value of 𝑏 is determined with equation(18.8). I consider this as an example; tetration to other values can becalculated in the similar way.
It is convenient to define 𝐵 = ln(𝑏). Then, the fixed points of logarithm
to base 𝑏, id est, solutions 𝐿 of equation ln𝑏 (𝐿) = 𝐿, can be expressed
through function filog, described in the previous section:

𝐿1 = filog(𝐵) ≈ 2.0565398441043761+1.1445267140098765 i (18.9)
𝐿2 = filog(𝐵 * )* ≈ 2.2284359658711805−1.3507994961102865 i (18.10)249

https://mizugadro.mydns.jp/t/index.php/File:Tetsheldonmap03.png
𝑦𝑢 + i𝑣 ≈ 2.0565 + 1.1445 i

8𝑣 = 1.26

𝑢=2

𝑣 = 1.2

𝑢=2

−20

−15

−10

1

−25

𝑣=

−30

𝑣=

0 .8

0 .8

𝑣 = −1𝑣 = −1.4
𝑢 = 1.2

𝑢 + i𝑣 ≈ 2.2284 − 1.3508 i

−8

𝑣=

0 .8

1

𝑣 = −1.4

−6

𝑢=2

𝑣 =0

𝑣 = −1
𝑢 = 1.2

𝑣 = 1.2𝑣=

𝑢 = 1.2

−4

𝑢=2

1

𝑣=

0.8

0
−2

𝑣 = 1.2
𝑣=

0.8

1

𝑣=

1

𝑣=

2

𝑢=2

𝑣=

𝑣=

4

𝑣 = 1.2−502𝑣 = −1𝑣 = −1.4468𝑣 = −1

𝑢 = 1.2
𝑣 = −1.4
10
15
20

𝑥

25

https://mizugadro.mydns.jp/t/index.php/File:Shelr80.png

𝑦
6
5
4
3
2
1

−1
−1
−2
−3
−4
−5

)︀
)
(𝑥

(︀ et 𝑏ℜ t=
𝑦 = tet1.5(𝑥)
𝑦 (︀
)︀
)
𝑥
𝑦 = ℑ tet𝑏(
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 𝑥

https://mizugadro.mydns.jp/t/index.php/File:Shelima600.png(︀)︀

𝑦 = ℜ tet𝑏 (i𝑥)

−8

−6

−4

(︀
)︀
𝑦 = ℑ tet𝑏 (i𝑥)

−2

𝑦
2
1
0
0
−1
−2

2

4

6

Fig. 18.3: Tetration to base 𝑏 = 1.52598338517+0.0178411853321 i :map: 𝑢+i𝑣 = tet𝑏 (𝑥+i𝑦) ;Intermediate plot: 𝑦 = ℑ(tet𝑏 (𝑥)) and 𝑦 = ℜ(tet𝑏 (𝑥)), thick curves;
the thin curve shows 𝑦 = tet1.5 (𝑥) from figure 17.1 ;Bottom plot: 𝑦 = ℑ(tet𝑏 (i𝑥)) and 𝑦 = ℜ(tet𝑏 (i𝑥))

250

8

𝑥

Top

where constant 𝐵 is evaluated as follows:

𝐵 = ln(𝑏) ≈ 0.4227073870410604 + 0.0116910660021443i (18.11)
Solution 𝐹 of the transfer equation 𝑏𝐹 (𝑧) = 𝐹 (𝑧 +1) with asymptotics

𝐹 (𝑧) = 𝐿1 +exp(𝑘1 𝑧 +𝜑1 ) + 𝒪(exp(2𝑘1 𝑧)) at ℑ(𝑧) → ∞𝐹 (𝑧) = 𝐿2 +exp(𝑘2 𝑧 +𝜑2 ) + 𝒪(exp(2𝑘2 𝑧)) at ℑ(𝑧) → −∞(18.12)(18.13)can be considered in the similar way, as for tetration to real base, larger
than the Henryk base 𝜂 = exp(1/e). Even the same contour of integration can be used. Subslitution of the asymptotic solutions (18.12),(18.13) into the transfer equation determines the increments

𝑘1 = ln(𝐿1 𝐵) ≈ −0.0047589243931785+0.5354935770338939 i (18.14)
𝑘2 = ln(𝐿2 𝐵) ≈ 0.0970758595007548−0.517289596155984 i (18.15)
The solution has quasiperiod

𝑃1 =

2𝜋i
≈ 11.7325200133916496−0.1042667514229599 i
𝑘1(18.16)in the upper part of the complex plane, and quasiperiod

𝑃2 =

−2𝜋i
≈ 11.7331504449085493−2.2018723603861230 i
𝑘2(18.17)in the lower part of the complex plane,
The properties above are sufficient to express the solution 𝐹 (𝑧) of thetransfer equation(︁)︁𝐹 (𝑧 +1) = exp 𝐵 𝐹 (𝑧)(18.18)through the Cauchy integral. This construction is quite analogous tothat for the tetration to base e, described in chapter 14; therefore I do
not repeat here the description of the contour of integration nor theiterational procedure, that provides the approximations of the solution.
The tetration is expressed through the solution 𝐹 in the following form:

tet𝑏 (𝑧) = 𝐹 (𝑧1 + 𝑧)

(18.19)where 𝑧1 is solution of equation 𝐹 (𝑧1 ) = 1. Using equation (18.18), thesolution can be extended at least to the right hand side of the complex
plane. As for the left hand side, in the Second quadrant of the complex
plane, the branch points and the cutlines appear. These cutlines appear
at the use of the logarithmic function to extend the solution to thedirection of negative values of the real part of the argument.
251

Unfortunately, the most of cut lines do not fit the frame of map in figure
18.3, althogh I make it in the whole width of the page. The asymptotic(18.12) indicates, that these cuts are unavoidable, while the imaginary
part of quasi period 𝑃1 is negative. Each time, when the tetration takes
value zero, there is branch point at value of the argument, for unity
smaller. These cuts are unavoidable also for other tetrations, while
formula𝐹 (𝑧) = log𝑏 𝑚 (𝐹 (𝑧 +𝑚))(18.20)is used for some integer 𝑚 such that value 𝑧 + 𝑚 belongs to the strip{𝑧 ∈ C : |ℜ(𝑧)| ≤ 1/2}. In this sense, the tetration to the Sheldon baseis similar to tetration to the real base.

3

Shell-Thron region

For moderate values of argument 𝑧 , tetration tet𝑏 (𝑧) looks as a smoothfunction of base 𝑏. At large values of ℜ(𝑧), dependently on the base,tetration either has complicated, quasi-chaotic behaviour, or approaches
some of the fixed point of the corresponding logarithm. At the site “Eretrande”, the range of values of base 𝑏, for which the tetration approaches
its limiting value, is called Shell-Thron region 1 . Such a name seems to
be commonly accepted, and I even suggest the Russian literal translation “Область Тронной Ракушки” for the Russian version of this Book.
However, yet, it is difficult to predict, how convenient and stable arethese names.
In this Book, I am interested mainly the tools for evaluation of superfunctions, and resolving the paradoxes, that are discussed among colleagues. Consideration of many examples, that cause no doubts, fall outof scope of this Book.
In principle, at the iterates of exponent to complex base, we have todeal with the 6-dimensional space. Coordinates of this space are thereal and imaginary parts of base 𝑏 (or real and imaginary parts of its
logarithm, 𝐵 = ln(𝑏)), the real and imaginary parts of the argument,andthe real and imaginary parts of the number of iterate. There may be
many interesting effects hidden in the 6-dimensional space. The detailed
description of these effects may require a special book, dedicated namely
1 Shell-Thron region:

https://math.eretrandre.org/hyperops_wiki/index.php?title=Shell-Thron_region.

252

to tetration to complex base. Since year 2010, Henryk Trappmann tries
to compose such a book [67] (I had promised to provide all the algorithms
and pictures he needs for this, and all the pictures requested up to dateare already plotted an included there). I hope, one day Henryk gets good
grant for this activity and will be able to polish the text to the state
he considers as satisfactory. While I still see no principal problem, thatcause serious difficulty at the evaluation of the corresponding iterates;
so, I do not include parts of [67] to this Book. For this reason, analysisof the Shell-Thron region is presented here in a declarative form.
I hope, the Readers can plot all the pictures that are necessary forillustration of tetration to the complex base, using the tools describedabove. However, if any difficulties of paradoxes appear in the analysis, I
shall try to consider and to resolve them. This point of view is describedby the Russian writers Arkadi & Boris Strugatski in the novel “Monday
begins on saturday” [14]: .. It’s nonsense to look for a solution if italready exists. We are talking about how to deal with a problem that has
no solution. ..

I hope, the Readers can evaluate tetration to other bases by themselves,using the tools any examples above. Following the idea mentioned,
I continue to deal with cases that are believed to have “no solution”.
One of the such cases refer to the superfunction of tetration; let it becalled “pentation”. In order to bring is to the textbook case (see figure18.4), this function, among other ackermanns, is considered in the next
chapter.

Fig. 18.4:

Textbook case253

Ackermanns

Fig. 19.1

W.Ackermann— One man asks horrible questions!
— Who is that man?
— He is called Asker-man!

https://mizugadro.mydns.jp/t/index.php/File:Ackermann.jpg

Chapter 19

Here I retell some results for holomorphic ackermanns, tetration andpentation [90]. For base 𝑏 > 1, tetration 𝑓 = tet𝑏 is solution of equations

𝑓 (𝑧 +1) = 𝑏𝑓 (𝑧) ,𝑓 (0) = 1

(19.1)

holomorphic at least in range𝑧 ∈ C : ℜ(𝑧) > −2

(19.2)𝑧 ∈ C : |ℜ(𝑧)| ≤ 1(19.3)

that is limited in range

Equation (19.1) appears as special case of the Ackermann equations

𝐴1 (𝑧) = 𝑏 + 𝑧 , 𝑧 ∈ C
𝐴𝑛 (1) = 𝑏 , 𝑛 ∈ N, 𝑛 ≥ 1(︀)︀𝐴𝑛 (𝑧 +1) = 𝐴𝑛−1 𝐴𝑛 (𝑧) , 𝑛 ∈ N, 𝑛 > 1(19.4)(19.5)(19.6)

I call functions 𝐴 “ackermanns” in order to avoid confusion with mathematician Wilhelm Ackermann, shown in figure 19.1. His last name, tomake difference from the name of the function, is written with Capitalletter [7]. Ackermanns 𝐴 are subject of this chapter. However, for thehighest ackernanns, the range of holomorphism has no need to be thesame as condition (19.2) for tetration. This range, as well as the asymptotic behaviour of highest ackermanns should be specified. I cannot yet
provide the general specification; below, I suggest only the first approachto the problem.
For base 𝑏 = 2, the explicit plots of the first four ackermanna are shownin figure 19.2 with thick lines, solid, solid, dotted and dashed. The thinlines refers to another (and more conventional) system of notations,described in the next section.
254𝑦 = tet (
2 𝑥)
𝑦=
2𝑥

𝑦=

−2
−1
2,

3
𝐴

1(

𝑥+

3)
−

3

4

𝒜
(1
,𝑥
)=

0
0

255

𝒜(
2,

1

,2 ( 𝑥

𝑦=

2

−1

−2

https://mizugadro.mydns.jp/t/index.php/File:Acker2t400.jpg

Fig. 19.2: Comparison of definitions for binary ackermanns

2) = tet (𝑥

3

𝑦 =𝐴
𝐴2
2,3 (𝑥
)=2𝑥
,2 (𝑥
)=
2𝑥

,4

+3
)−

𝑦 = 𝐴2 (𝑥

𝐴2

𝑦=

𝑥) =

)−3

2,3 (𝑥 + 3

6𝑦=, 𝑥) = 𝐴5
𝑦 = 𝒜(32,4 (𝑥 + 3) − 3

𝑦 = 𝒜(4, 𝑥) = 𝐴

)

𝑦

,

𝐴2

)
(1 𝑥
=

2
e

1

𝑦 = 2𝑥

𝑥

2+

𝑥

1

Binary ackermanns

In century 20, the ackermanns are considered mainly (or even exclusively) for the base 𝑏 = 2, and mainly (or even exclusively) for integer
values of the argument. I call ackermanns to base 𝑏 = 2 as “binary”
ackermanns.
For the binary ackermanns, the special notations are used. The numberof ackermann is written as additional, first argument, and the base 𝑏 is
not indicated at all; there is no need to indicate it, as it remains to be
two. Relation of the classical (and usual for today) notations for theackermanns by (19.4) (19.5) (19.6) can be expressed with the simple
formula𝒜(𝑚, 𝑧) = 𝐴2,𝑚 (𝑧 + 3) − 3(19.7)

In particular,

𝐴2,1 (𝑥) = 2 + 𝑥 = 𝒜(1, 𝑥+3) − 3 = 𝒜(1, 𝑥)
𝐴2,2 (𝑥) = 2 𝑥 = 𝒜(2, 𝑥+3) − 3
𝐴2,3 (𝑥) = exp2 (𝑥) = 𝒜(3, 𝑥+3) − 3 = 2𝑥
𝐴2,4 (𝑥) = tet2 (𝑥) = 𝒜(3, 𝑥+3) − 3(19.8)(19.9)(19.10)(19.11)Figure 19.2 shows relation between the single-argument function 𝐴 with
subscripts and conventional 𝒜 without subscript, but with two arguments. Four ackermanns are plotted as functions of real argument;𝑦 = 𝐴2,𝑚 (𝑥) for 𝑚 = 1, 2, 3, 4 are shown with thick lines (solid, solid,
dotted and dashed); dependences 𝑦 = 𝒜(𝑚, 𝑥) for 𝑚 = 1, 2, 3, 4 areshown with thin lines (solid, solid, dotted and dashed);
Due to relation (19.7), the thick lines in Fig.19.2 can be obtained fromthick lines by translation for 3 units along 𝑥 axis and for the same along
the ordinate axis; the only straight line for the First ackermann stays at
its place at such a transform.
Especially for the binary ackermanns, the system of equations (19.4),(19.5), (19.6) can be a little but extended, in order to include the “zeroth" ackermann. Equations for 𝒜 can be written as follows:

𝒜(0, 𝑧) = 𝑧 + 1
𝒜(𝑚+1, 0) = 𝒜(𝑚, 1)(︁)︁
𝒜(𝑚+1, 𝑧+1) = 𝒜 𝑚, 𝒜(𝑚+1, 𝑧)(19.12)(19.13)(19.14)

Displacemenr of both, argument and the function, by formula (19.7) canbe qualified as conjugation. The transfer equation (19.14) for the binary256

ackermann in the “classical” notations has the same form, as the transfer
equation (19.6).
One of applications of the ackermanns is to denote the huge numbers(for real moderate values of the argument). Due to the displacementof the argument, the canonical ackermanns can make illusion of a little bit more fast growth, than the ackermanns by (19.4), (19.5), (19.6).
However, for the applications, this “acceleration” is not important. I
think, the notations in formulas (19.4),(19.5),(19.6) are better than the
canonical notations. In notations (19.4), (19.5),(19.6), the third ackermann happens to be just exponent, the fourth - just tetration, and soon. In addition, I consider ackrmanns as superfunctions, holomorphic
with respect to the last argument; so, it is good, to keep this argument
single. The base 𝑏 and the number of the ackermann appear as parameters; they can be specified in the subscript. I hope, the Reader will meetno problems using relation (19.7) for the conversion from one system ofnotations to another one.

2

Names and notations

As I had mentioned above, in century 20, the functions of Ackermanns
are considered for base 𝑏 = 2 and only for integer values of the argument𝑧 . While, I see no fundamental limits, that would prohibit existence
and evaluation of these functions for various, including complex, values of 𝑏 and 𝑧 ; however, the appropriate requirement on the range ofholomorphism should be formulated.
I hope, for all ackermanns, we may require the real holomorphism at leastin some vicinity of at least some part to the real axis. In general, thesolution of the transfer equation is not unique; so, we should indicate
also the way of construction of each ackermann, or to guess (invent,postulate) its behaviour in the complex plane, following the idea, used
to construct natural tetration described in chapter 14.
Several ackermanns already have special names:
𝐴𝑏,1 = 𝑧 ↦→ 𝑏+𝑧 , addition of constant 𝑏,
𝐴𝑏,2 = 𝑧 ↦→ 𝑏𝑧 , multiplication by constant 𝑏,
𝐴𝑏,3 = exp𝑏 = 𝑧 ↦→ pow(𝑏, 𝑧) = 𝑧 ↦→ 𝑏𝑧 , exponent to base 𝑏,
𝐴𝑏,4 = tet𝑏 = 𝑧 ↦→ tet𝑏 (𝑧), tetration to base 𝑏,
𝐴𝑏,5 = pen𝑏 = 𝑧 ↦→ pen𝑏 (𝑧), pentation to base 𝑏.
The following functions can be denoted with symbols hex𝑏 , hep𝑏 , oct𝑏 ,..
and be called, accordingly, with terms “hexation”, “heptation”, “octation”
257

and so on. These notations appear at the truncation of Latin names ofnumbers to three characters.
Initially, the formalism of sulerfunctions had been developed for tetration, id est, for superfunction of the exponent, for the Fourth ackermann.
However, this formalism can be applied also for other functions, and, inparticular, for various ackermanns. As an example, in the next section
I consider the 5th ackermann, the pentation.

3

Pentation

If you give to some true Mathematician the teapot, the gas stove, thematches and water tap, and ask to prepare tea, the Mathematician puts
water into the teapot, ignites the gas and put the teapot on the fire. If
the Mathematician finds the tea powder, then, perhaps, he or she evendrop it into the hot water after to see the boiling and switch off the gas.
But if, after that, the Mathematician again is asked to prepare the tea,
while the water is already in the tea pot, and the gas fire at the stove isready.. Ooh.. You may guess, that the true Mathematician shuts down
the gas, drops the water from the teapot and spells the magic sentence:“The problem is reduced to the previous one!” .
I hope, with methods, described above, and keeping in mind the previous
paragraph, the Reader already can calculate the superfunctions, just following the general algorithms from this Book. On the other hand, I am
more physicists, than mathematician. (The mathematician colleagues
have no doubt in this, although some physicists express the opposite
opinion.) Therefore, instead of to spell the magic sentence “The problem is reduced to the previous one!” , in this chapter, I suggest one
more example of evaluation of ackermann. This example refers to thenatural pentation, id est, pen = pene = 𝐴e,5 .
At the construction of a superfunction, the key question is about the
fixed points of the transfer function. For pentation, the transfer function
is tetration, considered in chapter 14. The real fixed points of tetration
are shown in figure 19.3. This is modification of figure 17.1: some lines
are removed, and the new curve for tetration to base 𝑏 = 𝜏 ≈ 1.63532 is
added. Here, 𝜏 is specific values of base, at which the curve 𝑦 = tet𝜏 (𝑥)touches the line 𝑦 = 𝑥. The point of touching has coordinates (𝐿𝜏,1 , 𝐿𝜏,1 ),and 𝐿𝜏,1 ≈ 3.087. For this value, the additional grid lines are added infigure 19.3.
At base 𝑏 > 𝜏 , tetration tet𝑏 has the only one real fixed point. In
258

4

𝑏=

𝑏=e

1 .7
𝑏=
𝜏

𝑦

𝑏=
5
.
1
=
𝑏
𝑏 =√𝜂
𝑏= 2

𝐿𝜏,1
e

2

𝐿e,0

6
.
1

0
0

2

𝐿𝜏,1 4𝑥𝐿e,0
https://mizugadro.mydns.jp/t/index.php/File:Tet5loplot.jpg
Fig. 19.3: 𝑦 = tet𝑏 (𝑥), fragment of figure 17.1; line 𝑦 = 𝑥 and 𝑦 = tet𝜏 (𝑥) are added

particular, this refers to the case of natural tetration tet = tete . For
natural tetration, this point is 𝐿 = 𝐿e,0 ≈ −1.85035452902718, and forthis value, also the additional grid lines are shown in figure 19.3. Namely
this point is chosen to built-up the natural pentation pen, id est, for thefifth ackermann to base e = exp(1) ≈ 2.71 . As the Reader can see, notso many options we have in this case.
The growing real-holomorphic superfunction of natural tetration, that isconstructed with regular iteration at the fixed point 𝐿e,0 and approaches
this fixed point at −∞, I call the fifth ackermann, or pentation. Graphicof pentation is shown in figure 19.4. I describe the construction below.

259

𝑦

(𝑥 + 𝑥1 ))

6

en(𝑥)

𝑦 = 𝐿 + e xp ( 𝑘

5

𝑦 =p

4
3
e
2
1

𝑦 = 10 𝛿(𝑥)
−3

−2

−1

0

0

1

𝑥

−1
𝑦 = 𝐿e,4,0
https://mizugadro.mydns.jp/t/index.php/File:Penplot.jpg
Fig. 19.4: 𝑦 = pen(𝑥) by (19.24), its asymptotic (19.25), and deviation of the linear
approximation by (19.27)

260

For superfunction 𝐹 of transfer function tet, the transfer equation is(︁)︁
𝐹 (𝑧 +1) = tet 𝐹 (𝑧)(19.15)

I construct the growing along the real axis solution 𝐹 by the regular
iteration at the fixed point of tetration 𝐿 = 𝐿e,5,0 ≈ −1.85035452902718;
I mention the key point of the construction below.
For some natural number 𝑀 > 1, I search the solution 𝐹 of equation(19.15) in the following form:

𝐹 (𝑧) = 𝑓 (𝑧) + 𝑂(𝜀𝑀 )(19.16)where

𝑓 (𝑧) = 𝐿e,4,0 +

𝑀
−1
∑︁

𝑎𝑚 𝜀 𝑚(19.17)𝑚=1

𝜀 = exp(𝑘𝑧)(19.18)Here, the positive constant 𝑘 has sense of the increment of the growthof superfunction at large negative values of the argument, and 𝑎 are realcoefficients. For simplicity, I set 𝑎1 = 1. Substitution of representations(19.16), (19.17) to the transfer equation (19.15) and the asymptotic
analysis with small parameter 𝜀 give(︀)︀𝑘 = ln tet′ (𝐿) ≈ 1.865733(19.19)and the coefficients 𝑎; in particular,tet′′ (𝐿)/2)︁
𝑎2 = (︁≈ −0.6263241′′
tet (𝐿)−1 tet (𝐿)(19.20)tet′′ (𝐿)𝑎2 + tet′′′ (𝐿))︁𝑎3 = (︁≈ 0.4827′′2
tet (𝐿) −1 tet (𝐿)(19.21)For the numerical implementation, in (19.17), I choose 𝑀 = 4; this issufficient to evaluate pentation with 14 significant figures and to plot
all the figures of this article in real time. This approximation is good
for large negative values of the real part of argument of supertetration.
Then, for integer 𝑛, I define𝐹𝑛 (𝑧) = tet𝑛 (𝑓 (𝑧 − 𝑛))(19.22)The exact superfunction 𝐹 appears as limit

𝐹 (𝑧) = lim 𝐹𝑛 (𝑧)
𝑛→∞

261

(19.23)

𝑦

𝑣 =0

3

𝑢=𝐿
2

𝑣 = −1cut𝑣 =0−1𝑢 = 𝐿−2

𝑢=2𝑣 = −1

𝑣 =0

cut

𝑣 =1

𝑢=𝐿
−3

𝑢=1

𝑢 = −10 𝑣 =0

𝑢=0

𝑣 =1

1𝑢=𝐿

𝑣 =0

−4
−4

−3

−2

−1

0

1

2

3

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Penmap.jpg
Fig. 19.5:

𝑢+i𝑣 = pen(𝑥+i𝑦) по формуле (19.24)

The limit does not depend on the chosen number 𝑀 of terms in theasymptotic expansion. However, the larger is 𝑀 , the faster the limit in(19.23) does converge.
The pentation appears as superfunction 𝐹 with displaced argument:pen(𝑧) = 𝐴e,4 (𝑧) = 𝐹 (𝑥5 +𝑧)(19.24)

where 𝑥5 ≈ 2.24817 is solution of equation 𝐹 (𝑥5 ) = 1. Complex map of
this pentation is shown in figure 19.5.
The real-real plot of pentation by (19.24) is shown in figure 19.4 withthick curve. The additional horizontal grid line 𝑦 = 𝐿e,4,0 shows theasymptotic at large negative values of the argument. The thin curve262

shows the more advanced asymptotic

𝑦 = 𝐿e,4,0 + exp(𝑘(𝑥+𝑥5 ))(19.25)

Pentation is holomorphic at least for |ℜ(𝑧)| < |𝑃 |/2 ≈ 1.6838, where𝑃 = 2𝜋i/𝑘 is period; pentation, as exponential, is periodic function. A
little bit more than two periods are covered by the range of the map atfigure 19.5. Pentation has the countable set of cut lines, parallel to thereal axis. In figure 19.5, these cuts are marked with dashed lines.
In vicinity of the segment of length 2 at the negative part of the realaxis, pentation can be approximated with the linear function,pen(𝑥) ≈ 1 + 𝑥(19.26)At −2.1 < 𝑥 < 0.1, approximation (19.26) provides two significant figures. Deviation of this approximation from pentation pen can be expressed with function

𝛿(𝑥) = pen(𝑥) − (1+𝑥)(19.27)This deviation is shown in figure 19.4 with thin line; it is scaled withfactor 10; curve 𝑦 = 10 𝛿(𝑥) is drawn.
The linear function in the right hand side of equation (19.26) approximates also the previous ackermann, id est, tetration; its graphic is showin figure 14.1. For tetration, the function in the right hand side offormula (19.26) also gives of order of two significant figures; however,
the range of this approximation for pentation is twice wider, than for
tetration.
Complex map of pentation by (19.24) in figure 19.5 demonstrates, thatpen(𝑧) is holomorphic at least for ℜ(𝑧) < −2.5 . As the real part of theargument approaches minus infinity, pentation exponentially approaches
the limiting case 𝐿 = 𝐿e,4,0 ≈ −1.850354529, shown in figure 19.3. Inorder to show this explicitly, the light strip in figure 19.5 indicates the
additional level 𝑢 = 𝐿e,4,0 .
Pentation is periodic; its period 𝑃 is determined by the increment 𝑘 , idest, by the derivative of tetration at its fixed point 𝐿e,4,0 :

𝑃 =

2𝜋i
≈ 3.36767615657879 i𝑘
263(19.28)

𝑦
cut

𝐿
𝐿

𝑣 =1
𝑣=
−1

𝑣 =0

0

2

−2

𝑢=

𝑢=0

0

𝑢 = −1

1
2

2

1

𝑢=1

𝑃
2i
3
2

1

2

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Penzoo25t400.jpg
Fig. 19.6:

𝑢+i𝑣 = pen(𝑥+i𝑦) by (19.24), zoom-in from figure 19.5

The period is pure imaginary; the complex map of pentation reproduces
itself at translations for integer factor of the period. The cuts of therange of holomorphism are also reproduced.
Along the real axis, pentation shows very fast growth, faster, than thatof natural tetration. As maps of many other fast-growing functions, themap of pentation has complicated structure in vicinity of the real axis.
Pentation varies with huge derivatives, that correspond to enormous
density of the levels at the complex map. The plotter could not draw
the huge amount of lines, and so, the narrow region in vicinity of the real
axis in figure 19.5 is left empty. The same applies to the translations for
integer factor of period 𝑃 . In order to shown behaviour of pentation in
vicinity of the real axis and in vicinity of the cut line, figure 19.6 shows
the zoom-in from the figure 19.5.
General methods of construction of superfunctions can be used to buildup the ackermanns. If the growth of tetration happens to be not sufficiently fast, the pentation described above, can be used. The highest
ackermanns can be constructed in the similar way. In particular, thefixed points of pentation indicate the way to build-up its superfunction,
id est, the next ackermann. Fixed points of tetration are considered in
the next section.

264

4

Fixed poins of pentation and future work

In this section I suggest some hint, how can one built-up the next ackermann, hexation, using the fixed points of pentation. Actually, here I
do not construct hexation (because, anyway, I have to stop somewhere);
I only mention the way to do it.
As usually, one should begin with the fixed points, id est, solutions 𝐿 ofpen(𝐿) = 𝐿(19.29)Some of the solutions are shown in figure 19.6, these solutions are marked
with character 𝐿. They correspond to

𝐿 = 𝐿e,5,0 ≈ −2.260 + 1.384 i(19.30)

𝐿 = 𝐿e,5,1 ≈ 1.057 + 1.546 i(19.31)

There are also solutions in vicinity of the real axis

𝐿 = 𝐿e,5,2 ≈ 3.43 + 0.07 i(19.32)

𝐿 = 𝐿e,5,3 ≈ 4.39 + 0.11 i(19.33)but they do not fit the frame of the map on figure 19.6. The readers
are invited to solve numerically equation (19.29) and to adjust values of𝐿e,5,0 𝐿e,5,1 . I expect, one can find the real-holomorphic solution 𝐹 toequation(︁)︁𝐹 (𝑧 +1) = pen 𝐹 (𝑧)(19.34)

with additional conditions

𝐹 (0) = 1 , 𝐹 (i∞) = 𝐿e,5,0 ,(19.35)

The readers are invited to find this solution and interpret it as hexation,
id est, the 6th ackermann.
At this point, I stop constructing new ackermanns. I want to compare
the first 5 ackermanns. It is mater of the next section.

265

5

Comparison of natural ackermanns

5

e+
𝑦=

𝑦 = pen(𝑥)𝑦 = tet𝑦 = ex (𝑥)
p(𝑥)
𝑦=
e𝑥

𝑦

𝑥

In this section I compare the first 5 ackermanns for the natural base𝑏 = e. In such a way, I overview results for ackermanns. For real valuesof argument, these functions are plotted in figure 19.7.

4
3
e
2
1
(𝑥)
𝑦 = exp

0

1

2

e𝑥

−1
𝑦=

n

𝑦 =t

𝑦

e
=p

( 𝑥)

0

−1
et(𝑥)

−2

𝑦 = 𝐿e,4,0

https://mizugadro.mydns.jp/t/index.php/File:Ackerplot400.jpg
Fig. 19.7: 𝑦 = 𝐴e,𝑛 (𝑥) for 𝑛 = 1, 2, 3, 4, 5

266

𝑥

Functions plotted in figure 19.7 are:𝐴e,1 (𝑧) = e+𝑧 , addition of constant e,
𝐴e,2 (𝑧) = e 𝑧 , multiplication to constant e,
𝐴e,3 (𝑧) = exp(𝑧) = e𝑧 , natural exponent ,𝐴e,4 (𝑧) = tet(𝑧), tetration to base e,𝐴e,5 (𝑧) = pen(𝑧), pentation to base e.
These notations are used to mark curves 𝑦 = 𝐴e,𝑛 (𝑥) at figure 19.7. As
usually, if the base is not specified at the subscript, it is assumed to bee, base of natural exponent and that of natural logarithm.
Tetration and pentation at the segment [−1, 1] look similar. However,at the printing with good resolution, the deviation is seen, it exceeds
the width of the lines in figure 19.7.
I expect, the tetration already has the growth fast enough to describe
the fastest function that may appear in the applications. However, themain property of the scientific revolution is that they are unexpected. If
for some case, the growth of tetration is not fast enough, the pentation
or even higher sckermanns can be implemented, using the tools fromthis Book.
When I plotted the pentation, I ask myself: "Why not to do in thesimilar way the next ackermann?" The answer is simple: I already know,
how to do it. And the Reader, if reached here, also knows. Anyway, theBook should be finished at some moment, see figure 19.8.
Instead of to add more examples for the tools described above, I think,
it is more important to consider at least one example, when the tools
above do not work. This example correspond to the transfer function,that has no fixed points at all - neither real, nor even complex. Suchan example is considered in the next chapter, and, while typing this, I
believe, that will be the last example in this Book.
https://mizugadro.mydns.jp/t/index.php/File:Veslo.png
No rafting! Sit and finish your divineBook “Superfunctions”

Fig. 19.8: Wash yourself and finish your divine opera Khovanshina! [15]267Chapter 20
Without fixed points
In this chapter I describe the last (for this book) example of the transfer function, I consider function without fixed points. I suggest, that
the Reader tries to invent, to guess, to write-out some entire function
without fixed points, before to look at the formula below.

𝑦1Trappmann function

4
1+e

I give name ‘Trappmann function” to the elementary function

tra(𝑧) = exp(𝑧) + 𝑧

3e(20.1)

Function tra has no fixed point;
equation tra(𝐿) = 𝐿 has no solution. Function tra is compared toexponent in figure 20.1.

2
1
𝑦 = exp(𝑥)
−2

0
0

1
Function tra had been expected
to be a trap, trump, to catch−1
me on my claim, that I canbuild-up a superfunction for any−2growing real-holomorphic trans𝑦 = tra(𝑥)fer function. Henryk Trappmann
had suggested this function; so, I
20.1: 𝑦 = tra(𝑥) and 𝑦 = exp(𝑥)use first three caracters of his familyFig.
name
to denote it.
268−1

https://mizugadro.mydns.jp/t/index.php/File:TraplotT.jpg

For the transfer function without fixed points, neither method of regular
iteration, nor the representation through the Cauchy integral can beapplied as is for the primary evaluation of superfunction. For this reason,
this example is interesting and deserves the special chapter. Here, I retellresults published recently [89].

𝑥

𝑦

3

𝑣 =3

2

𝑣 =2

1

𝑣 =1

0

𝑣 =0

−1

𝑣 = −1𝑣 = − 16
𝑢 = 16−2𝑣 = −216𝑢=−−3𝑣 = −3−4𝑣 = −4−5𝑣 = −5−6−6

𝑢 = −1
𝑢=0
𝑢=1

𝑣 =4

𝑢 = −24

𝑢 = −3𝑣 =5𝑢 = −4

5

𝑣 =1

−5
−4
−3
−2
−1
0
1
2
3
4
https://mizugadro.mydns.jp/t/index.php/File:TraMapT.jpg
Fig. 20.2:

6

5

𝑥

𝑢+i𝑣 = tra(𝑥+i𝑦) по формуле (20.1)

Word “trap” can be interpreted also as gate, a way to the future successes
in solving of various transfer equations and iterates of tricky functions.
We see, function tra deserves the special name.
Complex map of function tra by (20.1) is shown in figure 20.2. In theright hand side of the figure, the map looks similar to that go exponent:
at the background of the exponential growth, the linear addition in theright hand side of equation (20.1) does not look impressive. In the left
hand side of the map, contrary, the exponent becomes negligible, andthe lines of constant real part and those of constant imaginary part of
the function form almost uniform grid of lines, parallel to the coordinate
axes.
Similar property, at least in some part of the complex plane, is shown by
the inverse function, id est, ArcTra = tra−1 . Complex map of ArcTra
is shown in figure 20.3. I consider its evaluation in the next section.
269

𝑦
8
6

𝑢=2

𝑢=0

𝑢 = −6𝑢 = −5𝑢 = −4𝑢 = −3𝑢 = −2𝑢 = −1𝑣 =3
2 𝑣 =2𝑣 =1
0 𝑣 =0𝑣 = −1−2 𝑣 = −2𝑣 = −3

𝑢=1

4

−4
−6
−8
−10

−8

−6

−4

−2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:ArcTraMapT.jpg
Fig. 20.3:

2

𝑢+i𝑣 = ArcTra(𝑥+i𝑦)

ArcTra, inverse of trappmann

In order to iterate a function, I mean non-integer iterates, first, we needto learn to evaluate the integer iterates. For positive integer number of
iterate, we may apply the function to the argument so many times asnecessary. As for the negative iterates, we need the efficient algorithm
for the inverse function. Let this inverse function be called ArcTra, in
analogy tithe ArcSin, ArcTan and ArcBessel; in some wide range tovalues of the argument (that includes the real axis), the ArcTra should
satisfy equation(︁)︁
tra ArcTra(𝑧) = 𝑧
Complex map of function ArcTra is shown in figure 20.3.
270(20.2)

File:LinearNonlinearOptic.png

Perhaps, I should explain, why I describe in this Book so elementary
thing, as building-up of inverse function. In century 20, I used to deal
with students interested in nonlinear
optics and quantum optics. Some ofthem were smart [28], but some students had problem even with linear
optics. It costed to me certain efforts,
to explain them, that there is a lot of
pretty “linear” science behind every socalled “nonlinear” phenomenon, as it Fig. 20.4: Linear versus nonlinear
is shown in figure 20.4. In the similarway, it is vain to discuss non-integer iterate, while even the negative
integer iterates are not yet implemented. I describe the implementationof function ArcTra in this section.

During the USSR, there was some science there. The famous institutesof the so-called "Soviet School" were Fizfak (Физфак, Physics department of the Moscow State University) and Fiztech (Физтех, Moscow
Phisics-Technical Institute). As one can guess, Fizfak used to deal with
fundamental science, and Fiztech did with the applied one. In order
to compare a specialist graduated from Fizfak to that graduated from
Fiztech, in the USSR, the following example is popular. One, graduatedfrom Fiztech, can calculate or ensemble everything. But he/she understands close to nothing. As for one graduated from Fizfak - Oooh.. heor she understands everything, although can calculate close to nothing.
I remind that story for the analogy with figures of this section. Readers,
who are interested in the beautiful pictures, may look at the complex
maps presented in this section. Then, these Readers can be qualified
with specification “understand everything”. As those graduated from
Fizfak.
For the implementation of the Abel function for the transfer function tra,the inverse function ArcTra = tra−1 is used. Unfortunately, as in thecase of ArcSin and ArcFactorial, I could not find any complex double implementation of ArcTra, and I had to make it by myself. This is general
rule: for the efficient implementation of the non-integer iterates of some271

transfer function 1 , the inverse function also should be implemented. Inorder to show the underwater stones, that may appear at calculation ofsuperfunctions, in this section, the efficient implementation of function
ArcTra is described.
It is not difficult to calculate the derivative of function tra:tra(𝑧) = 𝑧 + exp(𝑧) ,tra′ (𝑧) = 1 + exp(𝑧)(20.3)The Newton method gives a good precision evaluating solution 𝑓 ofequationtra(𝑓 ) = 𝑧

𝑧 − tra(𝑓𝑛 )tra′ (𝑓𝑛 )𝑓𝑛+1 = 𝑓𝑛 +𝑓 = lim 𝑓𝑛
𝑛→∞(20.4)(20.5)(20.6)

under a simple condition: the initial approximation 𝑓0 indicates the correct branch of the resulting inverse function. It is general rule, thatany non-trivial holomorphic function (and, especially, entire function)
has some points, where its derivative is zero; and these points provide
branches of the inverse function. For this reason, for the robust implementation, the correct choice of the initial approximation is essential: over-vice, the resulting function may return values from different
branches in some almost random, almost unpredictable way. In order
to indicate the correct branch, the expansions below are used.
Expansions of tra in vicinity of the saddle points ±𝜋i can be inverted,
giving the expansions of ArcTra in vicinity of −1±i𝜋 . These expansions
determine the positions of the branch points, and the direction of the
cut lines, seen in figure 20.3.
Consideration of the exponential in (20.3) as a small parameter andas a big parameter gives two more asymptotic expansions. Together
with the Taylor expansion at unity and the expansions at the branch
1 In principle, any holomorphic function can be declared as “transfer function”. I still specify

that tra is “transfer function”, in order to indicate my intention to iterate it and to distinguish itfrom its superfunction and its Abel function.
272

points, these expansions cover all the complex plane with the appropriate
primary approximations, providing several correct decimal digits of the
primary estimate 𝑓0 of value of function ArcTra. Then, few iterationsby (20.5) already reach the maximal precision (15 digits), available forthe complex double variables. This algorithm is used in the numerical
implementations of ArcTra and AuTra. 2
The easiest expansion is at the negative part of the real axis:

1
ArcTra(𝑧) ≈ app4 (𝑧) = 𝑧 − e𝑧 + e2𝑧 − e3𝑧2(20.7)The Readers are invited to plot the map of the agreement function(︂|tra(app4 (𝑧)) − 𝑧|
𝒜4 (𝑧) = − lg|tra(app4 (𝑧))| + |𝑧|)︂(20.8)The logarithmic growth of function ArcTra can be caught with theasymptotic expansion This expansion provides the approximation(︃𝑀

1 ∑︁ 𝑃𝑚 (ln(𝑧))ArcTra(𝑧) ≈ app3 (𝑧) = ln(𝑧) 1 +
𝑧 𝑚=0
𝑧𝑚

)︃
(20.9)

where 𝑃𝑚 is polynomial of 𝑚th order. In particular,

𝑃0 (ℓ) = −1

(20.10)𝑃1 (ℓ) =1 − ℓ/2(20.11)𝑃2 (ℓ) = −1 + 3ℓ/2 − ℓ2 /3(20.12)

1 − 3𝐿 + 11ℓ2 /6 − ℓ3 /4(20.13)𝑃4 (ℓ) = −1 + 5ℓ − 35ℓ2 /6 + 25ℓ3 /12 − ℓ4 /5(20.14)𝑃3 (ℓ) =The Taylor expansion of function tra at unity gives the approximation

ArcTra(𝑧) ≈ app1 (𝑧) =𝑀∑︁𝑐𝑛 (𝑧 − 1)𝑛(20.15)𝑛=1Some tens of coefficients 𝑐 of this expansion can be calculated, inverting the Taylor expansion of function tea at zero; the beginning of the
expansion can be written as follows:
2 Details of the algorithm and the maps of the primary approximations for ArcTra are loaded

to the special page https://mizugadro.mydns.jp/t/index.php/ArcTra .
273

2

3

4

5

6

7

8

𝑧
𝑧
𝑧
13𝑧
47𝑧
73𝑧
2447𝑧
ArcTra(1+𝑧) = 𝑧2 − 16
+ 192
+ 3072− 61440+ 1474560+ 41287680− 1321205760+..

At 𝑀 = 21, for |𝑧| < 1, this approximation provides at least 12 significant
figures.
Expansion of ArcTra at point −1 − i𝜋 can be written as follows:

ArcTra(𝑧) ≈ app2 (𝑧) =𝑀∑︁𝑐𝑚* (𝑧 +1+i𝜋)𝑚/2

𝑚=1

√ √
1
= −i𝜋 + i 2 𝑧 +i𝜋+1 + (𝑧 +i𝜋+1)33/2
i(𝑧 +i𝜋+1)2√−−(𝑧 +i𝜋+1)2 + ..
135
9 2

(20.16)

In more compact (and more efficient for the numerical implementation)
form, this expansion can be re-written as follows:

2𝑡2 2i𝑡3
8𝑡4
ArcTra − 1−i𝜋+2𝑡= −i𝜋 + 2i𝑡 +
−
−
3
9
135
6
7
5
32𝑡
139i𝑡
i𝑡
−
+
+ ..
(20.17)+
135 8505 42525
with obvious modification for the “conjugated” region
(︁
)︁
8𝑡4
2𝑡2 2i𝑡3
2+−
ArcTra − 1 + i𝜋 + 2𝑡
= i𝜋 − 2i𝑡 +
3
9
135
5
6
7
i𝑡
32𝑡
139i𝑡
−
−
−
+ .. (20.18)
135 8505 42525
where 𝑡 has sense of algebraic function of argument 𝑧 of the arctrappmann,√︂𝑧 + 1 − i𝜋𝑡=(20.19)2
The cut of the square root function in (20.19) automatically determines
the cut lines of function ArcTra, seen in figure 20.3. I invite the Reader
to plot the maps of the agreement functions(︂)︂
|tra(app𝑚 (𝑧)) − 𝑧|
𝒜𝑚 (𝑧) = − lg
(20.20)|tra(app𝑚 (𝑧))| + |𝑧|
(︀

2

)︀

in the complex plane 𝑧 = 𝑥 + i𝑦 for 𝑚 = 1, 2, 3, 4, id est, for the four
primary approximations suggested in this section.
When the approximations above are implementd and called with names
arctra1, arctra2, arctra3, arctra4, the numeral implementation of function ArcTra can follow the algorithm below:274

z_type arctran(z_type z) { DB x=Re(z), y=Im(z);if( x>2.) return arctra3(z);DB Y=fabs(y);if(Y<M_PI){ if(x<-1.5) return arctra4(z);if(Y<2.) return arctra1(z); }if( Y>5. || x<-4. ) return arctra3(z);if( y>0. ) return arctra2(z);return conj( arctra2(conj(z)) );}Alternatively, function ArcTra can be expressed through the Tania function (5.3):

ArcTra(𝑧) = 𝑧 − Tania(𝑧 −1)(20.21)

In such a way, in this Book, function Tania is used already 3 times
in three pretty different ways: First, in the simple model of the laser
amplifier, then, in the representation of function flog, and, en fin, now,
as an alternative implementation of ArcTra.
At least on vicinity of the real axis, function Tania can be expressed also
through other special functions namely, LambertW and WrightOmega,see (5.2). In principle, one could use those representations instead of theimplementation described above. However, at large 𝑧 , Tania shows thegrowth similar to the proportional; so, the numerical implementation of
ArcTra through Tania may cause loss of precision due to the rounding
errors. However, the special implementation of ArcTra specifid above,
had been used for the testing of the numerical implementation of function Tania.
While both tra and ArcTra are described and implemented, function
tra can be iterated, using the representation through the superfunction
and the abelfunction. I call these functions SuTra and AuTra; then, as
usually,(︁)︁
tra (𝑧) = SuTra 𝑛 + AuTra(𝑧)𝑛(20.22)In such a way I announce the future consideration: Function SuTra is
described in the next section, and function AuTra is treated soon afterthat section.
275

Supertrappmann

5
4

)

1+e

SuT
ra(𝑥

As it is mentioned above,
the Trappmann function tra
has no fixed points. For
this reason, the methods
described in the previous
chapters of this Book, cannot be applied “as is”. Henryk Trappmann had expected, that it will be difficult, to construct superfunction for the function
tra, if at all.

𝑦32𝑦 = −ln(−𝑥)−4 −3 −2 −1
−e

𝑦=

3

1
00

1

2

𝑥

−1

However, the simple and efficient way to construct andhttps://mizugadro.mydns.jp/t/index.php/File:SuTraPlo3T.jpg
evaluate SuTra, which is suFig. 20.5: 𝑦 = SuTra(𝑥) and 𝑦 = − ln(−𝑥)perfunction for the Trappmann function, exist. Plotof function SuTra is shown in figure 20.5 and compared to the graphicof its asymptotic. Below I describe the construction of this function.
Many colleagues, instead of to trace the deduction, ask first “How did you
guess?” Following such an interest, first, I describe the hint, that leads tothe efficient representation of function SuTra. For the transfer function
tra, and its superfunction 𝐹 , the transfer equation can be written as
follows:

𝐹 (𝑧 +1) = tra(𝐹 (𝑧)) = 𝐹 (𝑧) + exp(𝐹 (𝑧))(20.23)and re-written in the following form:

𝐹 (𝑧 +1) − 𝐹 (𝑧) = exp(𝐹 (𝑧))(20.24)In the left hand side of equation (20.24), I see something, that looks
similar to derivative of function 𝐹 . This similarity can be expressedwith approximate equation

d𝐹
1
≈
d𝑧
exp(−𝐹 )(20.25)that gives∫︁∫︁
exp(−𝐹 )d𝐹 ≈276

d𝑧(20.26)

(20.27)exp(−𝐹 ) ≈ −𝑧This relation indicates, that expansion of superfunction 𝐹 may begin
with the logarithmic term, namely the term, shown in figure 20.5.
However, the “heuristic solution” above is not a solution at all; in thebest case, it is only the asymptotic approximation. Substitution of theapproximation of function 𝐹 into the transfer equation (20.23) gives, ofcourse, some residual. This residual indicates the form of the next termof the expansion; and so on. In such a way, I guess the form of theasymptotic solution 𝐹 :(︂𝐹 (𝑧) = 𝐹𝑀 (𝑧) + 𝑂

ln(−𝑧)𝑧)︂𝑀 +1(20.28)where 𝑀 is natural number, and𝐹𝑀 (𝑧) = − ln(−𝑧) +

𝑀
∑︁

𝑃𝑚

(︁

)︁
ln(−𝑧) 𝑧 −𝑚

(20.29)

𝑚=1

where

𝑃𝑚 (𝑧) =𝑚∑︁

(20.30)𝑎𝑚,𝑛 𝑧 𝑛𝑛=1

Substituting this expansion into the transfer equation (20.23), I collectterms with equal powers of 𝑧 and equal powers of ln(−𝑧). This gives
both, verification of the form of the asymptotic expansion (20.29),(20.30)
and values of coefficients 𝑎. Several coefficients 𝑎, calculated in this way,
are shown in Table 20.1.
Table 20.1: Coefficients 𝑎 in the expansion (20.28),(20.29)

0

− 12

𝑎1,2

𝑎1,3

𝑎1,4

𝑎1,5

𝑎1,6

𝑎1,7

𝑎1,8

1
6
7
48
647
4320
1427
8640
13808637257600
327877314515200
251790467
914457600

− 14

1
8
3
16
5
16
25
48
5963
6912
97603
69120
1049251
460800

𝑎2,3

𝑎2,4

𝑎2,5

𝑎2,6

𝑎2,7

𝑎2,8

1
− 24

𝑎3,4

𝑎3,5

𝑎3,6

𝑎3,7

𝑎3,8

− 11
96

1
64
25
384
305
1536
537
1024
69953
55296

𝑎4,5

𝑎4,6

𝑎4,7

𝑎4,8

1
− 160

𝑎5,6

𝑎5,7

𝑎5,8

137
− 3840

1
384
49
2560
4123
46080

𝑎6,7

𝑎6,8

1
− 896

𝑎7,8

363
− 35840

1
2048

7
− 24

− 35
96
− 41638640− 18832880− 2171723
2419200− 35981749
29030400

− 17
64
653
− 1152− 39613456− 920881
414720263− 192013381− 30720

Using Mathematica, coefficients 𝑎 can be found with the following code:277

T[z_] = z + Exp[z];Clear [n, m, M];
P[m_, L_] := Sum[a[m, n] L^n, {n, 0, m}]; P[m, L];
F[z_]=-Log[-z]+a[1,1] Log[-z]/z+Sum[P[m,Log[-z]]/z^m,{m,2,M}]
M = 12;F1x = F[-1/x + 1];
Ftx = T[F[-1/x]];s[1] = Series[F1x - Ftx, {x, 0, 2}];t[1] = Extract[Solve [Coefficient[s[1], x^2] == 0, {a[1, 1]}], 1]A[1, 1] = ReplaceAll[a[1, 1], t[1]];su[1] = t[1]m = 2; s[m] = ReplaceAll[Series[F1x - Ftx, {x, 0, m + 1}], su[m]];t[m] = Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m + 1)];u[m] = Collect[t[m], L];v[m] = Table[Coefficient[u[m] L, L^(n + 1)] == 0, {n, 0, m}];w[m] = Table[a[m, n], {n, 0, m}];ad[m] = Extract[Solve[v[m], w[m]], 1];su[m + 1] = Join[su[m], ad[m]];ReplaceAll[ReplaceAll[F[x], su[m + 1]], Log[-x] -> L]
m = 3; s[m] = ReplaceAll[Series[F1x - Ftx, {x, 0, m + 1}], su[m]];t[m] = Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m + 1)];u[m] = Collect[t[m], L];v[m] = Table[Coefficient[u[m] L, L^(n + 1)] == 0, {n, 0, m}];w[m] = Table[a[m, n], {n, 0, m}];ad[m] = Extract[Solve[v[m], w[m]], 1];su[m + 1] = Join[su[m], ad[m]];ReplaceAll[ReplaceAll[F[x], su[m + 1]], Log[-x] -> L]and so on 𝑚 = 4, 𝑚 = 5, etc. I do not arrange here the loop “For” withrespect to 𝑚, in order to keep the code explicit and simplify the tracing
step by step.
Expression (20.29) can be considered as primary approximation of superfunction of the Trappmann function (20.1). Then, the exact solution
𝐹 of the transfer equation appears as limit(︀)︀
𝐹 (𝑧) = lim tra𝑘 𝐹𝑀 (𝑧 −𝑘)𝑘→∞(20.31)

In order to get superfunction SuTra, that satisfies also the additional
condition

SuTra(0) = 0(20.32)

SuTra(𝑧) = 𝐹 (𝑧 + 𝑥0 )(20.33)

I set

278

𝑦

−2
.2
𝑢=
𝑢 = −2𝑢 = −1.8
𝑢 = −1
𝑢 = −1 .6
𝑢 = − 1 .4
−1 .2

2

𝑢=

1

4

𝑣=

6

𝑣=

2

8

𝑣 =3

𝑣 = 0 .2𝑣 =00𝑣 = −0.

−22𝑣 = −3

−2

𝑣=

−6

𝑣=

−1

−4

−8
−8

−6

−4

−2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:SuTraMapT.jpg
Fig. 20.6: 𝑢+i𝑣 = SuTra(𝑥+i𝑦) по формуле (20.33)

where 𝑥0 ≈ −1.1259817765745026 is real solution of equation

𝐹 (𝑥0 ) = 0(20.34)

Figure 20.5 shows 𝑦 = SuTra(𝑥) versus 𝑥. For comparison, the thincurve shows the leafing term of the asymptotic representation of SuTra,
id est, 𝑦 = − ln(−𝑥). In the left hand side of the figure, graphic of
SuTra approaches its asymptotic.
Complex map of function SuTra is shown in figure 20.6. Being far fromthe positive part of the real axis, function SuTra looks similar to function𝑧 → − ln(−𝑧), as it is suggested by the leading term of its asymptotic
representation. Lines of the constant real part look similar to circles,
while lines of constant imaginary part look similar to the straight lines;
this make the map to look similar to the map of logarithm.
279

𝑦

8

8

𝑣 =0

−2

−2
−1
.8
𝑢=

𝑣 =0
𝑣 = −0.2

−2

𝑣=

𝑣=

−8

−2

−2

−6

𝑣=

𝑣=

−6

−1

−4
−1

−4

𝑣=

−2
.2

𝑣 = 0.20𝑣 = −0.2

𝑢=

𝑣=

2

𝑣 = 0.2

0

𝑢=

−2
−1
.8
𝑢=

𝑢=

𝑢=

2

1

4

𝑣=

6

1

4

𝑣=

6

−2
.2

2

2

𝑦

−8
−8

−6

−4

−2

0

2

4

6

8

𝑥

−8

−6

−4

−2

0

2

4

6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sutralomap.jpg
Fig. 20.7:

Complex maps of functions 𝑓2 by (20.35) and 𝑓4 (20.36), over-lapped with map of function 𝑓∞ by (20.37)
I would like to show, how the logarithmic function 𝑧 → − ln(−𝑧) canbe approximated with the entire function. Figure 20.7 shows complex
maps of the entire functions

𝑓2 (𝑧) = SuTra(2𝑧) + ln(2)(20.35)𝑓4 (𝑧) = SuTra(4𝑧) + ln(4)(20.36)for comparison, each of these maps is overlapped with the map of𝑓∞ (𝑧) = − ln(−𝑧)(20.37)In the right hand side picture of figure (20.7), the levels of different
functions are so close, that it is difficult to see, which of functions does
each level corresponds to. For all 𝑧 , except zero and positive part of thereal axis, sequence(︁)︁Φ𝑛 (𝑧) = − SuTra(−𝑛𝑧) + ln(𝑛)(20.38)at big 𝑛 approximates ln(𝑧). Up to my knowledge, expression (20.38)
provides the range of approximation of logarithm with entire functions,that is wider than that of all approximations ever suggested before publication [89].
I have no idea, what for the approximation of logarithm with entire
function can be used. This approximation appears here as a by-productat construction of function SuTra. But if someone needs such an approximation, it is done and it is here, formula (20.38).
280

4Implementation of SuTra and the testing

The key parameters of any approximation are the range of applicability, the precision and the number of elementary operations that shouldbe performed for each evaluation. If the range of applicability is wide(for example, cover the range of holomorphism of the approximatedfunction), the precision is close to the that allowed at the computer representation of numbers, and the complex maps can be plotted in realtime, then I qualify the solution as “exact”. This implies, that, if, forsome reason, the precision needs to be improved, the implementation of
the high precision code is straightforward, id est, can be realised withthe same formulas.
If some function is supplied with definition, the properties are described,
and the efficient precise algorithm for the evaluation is presented, I treatsuch a function as a “special function”. Then, if the solution of someproblem is explicitly expressed in terms of the special functions, I call it“exact solution”. I place this explanation here form my old coauthor, who
believed, that even 𝜋 is approximate number. However, the fundamental mathematical constants are, contrary, exact numbers, in the sense
mentioned above; neither 𝜋 , nor values of superfunctions constrcted inthis Book are exceptions: they are exact and can be evaluated with any
required precision.
Following the ideology of the preamble of this section, I consider here therange of values of the argument 𝑧 , at which the approximation 𝐹𝑀 (𝑧+𝑥0 )by (20.29) can be considered as good approximation of SuTra(𝑧). Then
I describe the algorithm, based on this analysis.
For the practical reasons (in order to get the complex double implementation, that can be easily reproduced and verified), I took 12 terms in
the expansion, id est, 𝑀 = 12. For the dozen terms, the agreement(︂|𝐹𝑀 (𝑧 +𝑥0 ) − SuTra(𝑧)|𝒜(𝑧) = − lg
|𝐹𝑀 (𝑧 +𝑥0 )| + |SuTra(𝑧)|)︂(20.39)had been analysed. Levels of this function are shown in figure 20.8.
For the precision complex double, values outside the “thick” contour infigure 20.8 can be used “as is”. For other values, formula
281

𝑦
18

Range used for approximation of SuTra

𝒜 = 15𝒜 = 14𝒜 = 13
𝒜 = 12

16
14
12

𝒜 > 15

10
8
6

𝒜=9

4
2
0

−18 −16 −14 −12 −10−8 −6 −4 −2

0

2

4

6

8

10

12

14

16

18

20

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Sutraamap.jpg
Fig. 20.8: Map of 𝒜 = 𝒜(𝑥+i𝑦) by (20.39)

SuTra(𝑧) = tra𝑘 (SuTra(𝑧 −𝑘))(20.40)is used with appropriate 𝑘 , in such a way, that the inner argument is
outside the “thick” contour in figure 20.8. This thick contour is not partof the map; it is formed with the segment along the line 𝑥 = −11, arc
with centre at point (5, 0) of radius 18, and the half-line along 𝑦 = 6.
Outside the thick contour in figure 20.8, the precision of evaluation offunction SuTra is limited mainly by the rounding errors. Values outside
the “thick contour” are used for the numerical implementation directly.
For other values, formula 20.40 is used with appropriate value of 𝑘 . Thisalgorithm is used for the numerical implementation of function SuTra,

https://mizugadro.mydns.jp/t/index.php/Sutran.cin
At the evaluation of function SuTra, function tra should be evaluated oforder of ten times. (However, this depend on the initial value). Function
tra is fast, because tra(𝑧) = 𝑧+exp(𝑧). In such a way, the evaluation of
SuTra is only for an order of magnitude slower, than evaluation of otherspecial functions like exp, erfc or BesselJ. This is one of reasons why I
qualify SuTra as special function.
For iterates of the Trappmann function, I need also the inverse function,

AuTra = SuTra − 1. This function is shown in figure 20.9 and describedin the next section.
282

−𝑥

𝑦

1

0

0

−1

𝑦=

+
𝑐0

e
−
2
𝑥/

𝑦=

1

𝑥
(
a
r
AuT

2

)

𝑥

𝑦 = 𝑐0 +𝑥/2−e−𝑥 −e𝑥 /6

https://mizugadro.mydns.jp/t/index.php/File:Autraplot.jpg
Fig. 20.9: 𝑦 = AuTra(𝑥) by (20.47) and two its asymptotics by (20.44)

5

AuTra, abelfunction of trappmann

This section describes evaluation of the inverse function of SuTra, id est,
the Abel function for the Trappmann function tra. I call this abelfunction AuTra. Its explicit plot is shown in figure 20.9; for comparison, inthe same picture, the two asymptotics of function AuTra are shown.
Complex map of function AuTra is shown in figure 20.11.
For the transfer function tra, the abelfunction 𝐺 satisfies the Abel equation

𝐺(tra(𝑧)) = 𝐺(𝑧) + 1(20.41)

In order to see the asymptotic expansion of the solution 𝐺, I rewrite283

this equation as

𝐺(𝑧 + e𝑧 ) − 𝐺(𝑧) = 1(20.42)and expand the left hand side, using e𝑧 as small parameter:𝐺′ (𝑧)e𝑧 + 𝐺′′ (𝑧)𝑒2𝑧 /2 + .. ≈ 1(20.43)This expansion allows to guess the asymptotic solution:𝑧e𝑧 e2𝑧 19e3𝑧 e4𝑧 41e5𝑧−𝑧
𝐺(𝑧) ≈ − e − +−+−+ 𝒪(e6𝑧 ) (20.44)
2
6
16
540
48
4200
The coefficients of the expansion above are calculated and evaluatedwith the Mathematica code below:
tra[z_]=z+Exp[z];g0[z_] = z/2 - Exp[-z] + Sum[c[n] Exp[n z], {n, 1, 20}]n = 1; s[n] = Series[g0[Log[t]] + 1 - g0[tra[Log[t]]], {t,0,n+1}]
u[n] = Extract[Solve[Coefficient[s[n], t^(n+1)] == 0, c[n]], 1]g[n, z_] = ReplaceAll[g0[z], u[n]]For[n = 1, n < 20, n++;s[n] = Series[ g[n-1,Log[t]]+1-g[n-1,tra[Log[t]]],{t,0,n+1}];
u[n] = Extract[Solve[Coefficient[s[n], t^(n+1)] == 0, c[n]],1];g[n,z_] = ReplaceAll[g[n-1, z], u[n]];]g[n, z]Table[Coefficient[g[n, z], Exp[n z]], {n, 1, 20}]
N[Table[Coefficient[g[n, z], Exp[n z]], {n, 1, 20}], 18]

The same coefficients can be obtained also by the inversion of the asymptotic expansion of function SuTra. Note, that the asymptotic expansionof AuTra is simpler, than that of SuTra.
For some fixed integer 𝑀 , define the primary approximation as truncation of the series above:
𝑀
∑︁
𝑧
−𝑧
𝑐𝑚 e𝑚𝑧
𝐺𝑀 (𝑧) = − e +2
𝑚=1(20.45)

Define function 𝐺 as limit

𝐺(𝑧) = lim (𝐺𝑀 (ArcTra𝑛 (𝑧)) + 𝑛)𝑛→∞(20.46)

AuTra can be expressed through 𝐺 with

AuTra(𝑧) = 𝐺(𝑧) − 𝐺(0) ≈ 𝐺(𝑧) + 1.1259817765745026(20.47)The constant 𝐺(0) can be interpreted also as negative of coefficient 𝑥0
in equation (20.34), id est, 𝑥0 = −𝐺(0).
284

𝑦

𝒜<1

3
2

𝒜<1

𝒜=3
𝒜=1

0

𝒜 = 12
𝒜=9
𝒜=6

𝒜 = 151−1−2
𝒜 ≈ 16

−3
−4

−3

−2

−1

0

𝑥

https://mizugadro.mydns.jp/t/index.php/File:Autran0m9tes64t.jpg
Fig. 20.10:

Map of agreement 𝒜 = 𝐴(𝑥+i𝑦) by (20.50); range (20.49) is

shaded 𝒜 = 𝐴(𝑥+i𝑦) по формуле (20.50)
For the numerical implementation, we need to choose the appropriate
number 𝑀 in and to determine the number 𝑛 of iterations approximating limit in equation (20.46). For the complex double arithmetics, thereasonable choice is 𝑀 = 9. Then, the primary approximation

SuTra(𝑥 + i𝑦) ≈ 𝑔9 (𝑥 + i𝑦)(20.48)is used for the region defined with condition|𝑦| < 3 and |𝑦|/3+𝑥 < 3.5(20.49)This region is shaded in the figure 20.10 Also, at the same figure, themap of agreement 𝐴 is shown,
285

(︂
)︂
|SuTra(𝑔𝑀 (𝑧))−𝑧|
𝒜(𝑧) = − lg
|SuTra(𝑔𝑀 (𝑧))|+|𝑧|(20.50)

Map in figure (20.10) can be considered also as the numerical verificationof relation

SuTra(AuTra(𝑧)) = 𝑧(20.51)In the shaded range, for the numerical implementation, the relation(20.51) holds with at least 15 decimal digits; as it is supposed to be
while AuTra is abelfunction, corresponding to superfunction SuTra forthe transfer function tra.
Region, where the primary approximation is used, can be optimised,
approaching to the level 𝒜 = 15 and improving the algorithm. I suggest,the Reader can do the as an exercise; I hope, the reader will not forget
to test the improved algorithm.
The primary approximation (20.48) can be used only in the narrow rangeof values of the argument shaded in figure autraAgreMap. If the initialargument 𝑧 = 𝑥 + i𝑦 happens to be outside the shaded region, then,the function ArcTra = Tra−1 is applied 𝑛 times with such 𝑛, that theargument comes to the shaded range. Then, approximation

AuTra(𝑧) ≈ 𝑔9 (ArcTra𝑛 (𝑧)) + 𝑛(20.52)is used to evaluate the function.
The cuts of the range of holomorphism of function ArcTra determine
also the cuts the range of holomorphism of function AuTra. These cuts
are seen in figures 20.3 and 20.11. At large |𝑧|, function AuTra(𝑧)shows the slow growth, except the half-strip ℜ(𝑧) < 0, |ℑ(𝑧)| < 𝜋 . In
this half-strip, at large negative values of ℜ(𝑧), function |AuTra(𝑧)|increases exponentially. In particular, this refers to the real values of
the argument. This behaviour agrees with that of SuTra shown in figure
20.5.
Readers are invited to download the generators of the figures of thischapter, together with implementations of SuTra and AuTra, and investigate numerically the ranges of applicability of identities

AuTra(SuTra(z)) = z , SuTra(AuTra(z)) = z(20.53)

With functions SuTra and AuTra, described in this chapter, one canevaluate the non-integer iterates of function tra. These iterates are described in the next section.
286

𝑦

𝑢=

𝑣=

8

0.6

𝑣=

0 .4

3

6
4

cut

𝑣=

0 .2

𝑣 =0

0

𝑢 = 2.4

𝑢=2
𝑢 = 2.2

2

−2

𝑣=

cut

−4

𝑣=

−10

−8

𝑢=

−8

−0
.6
−6

−4

−0
.2

2
𝑢=

3

−6

𝑣 =0

−2

0

2

4

6

.6

8

𝑥

https://mizugadro.mydns.jp/t/index.php/File:AuTraMapT.jpg
Fig. 20.11:

6

𝑢+i𝑣 = AuTra(𝑥+ i𝑦)

Iterates of trappmannWith functions SuTra and AuTra, the iterate of the Trappmann function
can be expressed as follows:(︁)︁
tra (𝑧) = SuTra 𝑛 + AuTra(𝑧)𝑛(20.54)

As usually, the number 𝑛 of iterate has no need to be integer. For realvalue of the argument and some real values of the number 𝑛, the iteratesof the Trappmann function are shown in figure 20.12.
Iterates of function tra(𝑧) = 𝑧+e𝑧 look similar to those of other growing
transfer functions. These iterates provide the smooth transfer from thefunction tra to its inverse function ArcTra = tra−1 , and zeroth iterateis the identity function.
287

𝑛=𝑛 = 0 .4
𝑛 = 0.
𝑛 = 0.23
𝑛= 0
.
0 1

1
𝑛=

𝑛 = 10
𝑛=8
𝑛=6
𝑛=5
𝑛=4
𝑛=3
𝑛=2

𝑦

.2
0
−
=
𝑛 0.4
−
=
𝑛

2
1
−2

−1

00

1
−
=
𝑛

1

−1

−2

2

𝑥
𝑛 = −2𝑛 = −3𝑛 = −4𝑛 = −5𝑛 = −6𝑛 = −8𝑛 = −10

https://mizugadro.mydns.jp/t/index.php/File:TraitT.jpg
Fig. 20.12:

7

𝑦 = tra𝑛 (𝑥) versus 𝑥 for various 𝑛 by (20.54)

Relation to other functions

Function SuTra can be expressed through function SuZex

SuTra(𝑧) = SuZex(ln(𝑧))(20.55)and function AuTra can be expressed through function AuZex

AuTra(𝑧) = exp(AuZex)(20.56)One can look at expansions of SuZex and AuZex from chapter 12 andsee, that expansions for functions SuTra and AuTra can be obtainedwith relations (20.55) and (20.56). However, these representations may
loss some precision, especially in the regions where the logarithm has low
derivative, and the exponential has high derivative. So, for the numericaltests, I use the special representations from the previous sections.
288

In order to make the deep test of relations, suggested in this Book, therobust representations are required for all the functions involved. For
these reasons, the special representations and implementations for functions ArcTra, SuTra and AuTra are described above. The readers areinvited to make the asymptotical analysis for the relations above. Also,
the numerical verification can be used. For moderate values of the argument, the representation of SuTra through SuZex loss only few decimal
digits in the precision. I load the descriptions of functions mentioned inthis section,

https://mizugadro.mydns.jp/t/index.php/ArcZex
https://mizugadro.mydns.jp/t/index.php/AuZex
https://mizugadro.mydns.jp/t/index.php/SuZex
https://mizugadro.mydns.jp/t/index.php/ArcTra
https://mizugadro.mydns.jp/t/index.php/SuTra
https://mizugadro.mydns.jp/t/index.php/AuTra,
together with their complex double numerical implementations.
On this I finish the consideration of the Trappmann function and its
iterates. And at this point I finish consideration of examples of transfer
functions, superfunctions, abelfunctions and the non-integer iterates.
Perhaps, I should explain, why I had spent so many efforts on this. I
think, the best explanation is to remind the old folkloric story below.
Оne-legged friend of one Taylor asked him to sew the special one legpants. He payed well for the custom pants, but he needed also the pants
for his dog, who, as himself, had lost one leg long time ago.Taylor sewed
the pants for that 3-leg dog. The pants were beautiful, and friend offriend asked him the same for his normal, 4-leg dog.. The story is long,the starfishеs and octopuses are mentioned there. En fin, the Taylor had
elaborated tools to sew pantaloons for creatures with arbitrary number
𝑛 of legs. And if tomorrow some extraterrestrials with 𝑛 legs come, theTaylor already has pantaloons for them.
I typed the story above in order to explain better, what is scientific research and what do the researchers. You may consider this as a kind ofjoke, but if someone needs to approximate, for example, the logarithmic
function with entire functions, - then, like the Taylor mentioned, I already have such an approximation; it can be expressed through SuTraby formula (20.38). This may be considered as a small specific addition
to various motivations suggested in the Preface.
On this point I stop the main body of the Book and go to overview theresults presented above. This makes the content of the next chapter.
289

Chapter 21
Conclusion
This chapter discusses the same, as the Introduction. The main difference of this chapter
from the introduction is, that I assume, thatthe reader already has browsed at least someFig. 21.1:of the previous chapters, as it is shown in figure
21.1, and understands sense of some notations
used. In order to show the need of conclusion, I remind the old legend.
One emperor wanted to study history. He ordered the Ministry of Science to develop a full course of the world-wide history. The greatest
scientists were working on this tutorial during many years. Finally, theheavy truck arrived with thousands volumes of "Complete Course of
the World History". The Emperor realised that all his life will not be
sufficient to read this course. The Emperor asker the President of the
Ministry of Science to shorten the course. The historians worked on thesecond edition during few years, and then, in a big pack, "Trilogy of theWorld History" had been delivered to the Emperor. But the Emperor
already had weak eyes, and he could not read that Trilogy. Again, thehistorians had to shorten the course. A year later, the Top Historist
came to the Emperor and gave him the pamphlet "A Brief History of
the Imperial Family." Emperor was old and ill, and could not even readthat brochure. He asked Historist, whether the brochure can be reduced.
Historist answered: “ No new edition is necessary. I’ll tell you right now:
People were born, suffered and died.”
Several colleagues had told me, that this Book is too thick, and asked,
if it can be shorter. As the historian in the story above, I follow therequests and describe the topic of superfunctions briefly. Below, themain results of the Book are collected in a single section.

290

1

Basic results

Holomorphic functions can be iterated. The iterates can be expressed
through the superfunction and the abelfunction; then, the number of
iterate has no need to be integer.
In order to iterate some holomorphic function 𝑇 , first, I declare it as“transfer function”. Then, I construct for the superfunction 𝐹 , that issolution of the transfer equation(︁)︁𝐹 (𝑧 +1) = 𝑇 𝐹 (𝑧)I need also the inverse function, 𝐺 = 𝐹 −1 . I call it “Abel function” or“abelfunction”. It satisfies the Abel equation(︁)︁
𝐺 𝑇 (𝑧) = 𝐺(𝑧) + 1
When superfunction 𝐹 and the Abel function 𝐺 are established, theiterates of function 𝑇 can be expressed as(︁)︁
𝑇 (𝑧) = 𝐹 𝑛 + 𝐺(𝑧)𝑛where the number 𝑛 of iterate has no need to be integer; the transfer
function can be iterated some non-integer, rational or even complex
number of times. However, for the integer 𝑛, the conventional expressionof iterates holds:

(︁ (︀
)︀)︁
𝑇 𝑇 ..𝑇 (𝑧)..
𝑇 𝑛 (𝑧) =⏟⏞𝑛 evaluations of function 𝑇
Solution of the transfer equation is not unique. If some solution 𝐹 isfound, then, one additional solution 𝑓 can be constructed with modification of the argument,(︁)︁
𝑓 (𝑧) = 𝐹 𝑧 + 𝜃(𝑧)where 𝜃 is periodic holomorphic function with period unity. Accordingly,
the new abelfunction 𝑔 can be established. Then, the new superfunction
and the corresponding abelfunction will provide new, different iteratesof the transfer function.
Variety of superfunctions can be narrowed, if we establish, postulate theasymptotic behaviour of the superfunction in the complex plane. The
291

superfunction with simple asymptotic behaviour I treated as principal.
Other superfunctions can be expressed with modification of argumentof the principal superfunction. Holomorphic periodic function grows
at least exponentialy (as sin and cosine to in the direction of imaginaryaxis); even a small periodic modification is easy to reveal in the complex
plane. For this reason, for uniqueness, it is important to build-up thesuperfunctions for the complex argument, even if they are supposed to
be used for the real argument. The criterion of holomorphism indicates,
which of superfunctions is expected to have the physical sense and should
be considered as “true” one.
Some superfunctions and abelfunctions have special names. These functions are collected in Table 3.1. Some of them are widely known; one
may use them without to know, that they are superfunctions.
In principle, for any holomorphic function 𝐹 , one can built-up the in(︀)︀verse function 𝐺 = 𝐹 −1 and define 𝑇 (𝑧) = 𝐹 1 + 𝐺(𝑧) . Then, suchfunction 𝑇 can be treated as transfer function with known superfunction(︀)︀𝐹 , abelfunction 𝐺 and non-integer iterates 𝑇 𝑛 (𝑧) = 𝐹 𝑛 + 𝐺(𝑧) .
The inverse problem, id est, construction of superfunction 𝐹 for somegiven transfer function 𝑇 , is considered in this Book. For this construction, the key question is about the fixed points of the transfer function,
id est, about solutions 𝐿 of the equation

𝐿 = 𝑇 (𝐿)As physicist, I am interested mainly in the real-holomorphic functions,for which 𝑇 (𝑧 * ) = 𝑇 (𝑧)* . In order to reduce variety of solutions of thetransfer equation, I postulate, that the superfunction approaches the
fixed point 𝐿 at the infinity.
Methods of asymptotic expansions of the superfunctions are suggested
in the Book. At infinity, the superfunction is postulated to approach
the fixed point of the transfer function. In many cases, this approachis exponential; and, in many cases, the exact superfunction appears
as the limit at the multiple application of the transfer function to theasymptotic solution with displaced argument.
It may happen, that all the fixed points 𝐿 of the transfer function 𝑇 arecomplex, not real. In particular, this is case of the natural exponent.
Then, the superfunction can be expressed through the Cauchy integral
and solution of the corresponding integral equation. Historically, the292

first complex map of superfunction of exponent, had been constructedwith this representation. Complex map of tetration is shown in figure
14.12 and at the First page of the cover of this Book. Tetration tet
appears as solution of the transfer equation

tet𝑏 (𝑧 +1) = etet𝑏 (𝑧) , tet𝑏 (0) = 1
tet𝑏 (𝑧) is supposed to be bounded in the strip |ℜ(𝑧)| ≤ 1. For real 𝑏 > 1
and real 𝑥, dependence 𝑦 = tet𝑏 (𝑥) is shown in figure 17.1. Tetration
can be constructed also for the complex base; the example with 𝑏 =1.52598338517+0.0178411853321 i is shown in figure 18.3.
It may happen, that the transfer function 𝑇 has no fixed point at all.
One example of such a function is the Trappmann function

𝑇 (𝑧) = tra(𝑧) = 𝑧 + exp(𝑧)However, even for this function, the superfunction can be constructed.
It is called SuTra; its map is shoe in figure 20.6. This is entire function
with logarithmic asymptotic; up to my knowledge, before publication[89], no one such function had been suggested.
Since 2010, I claim, that I can construct the superfunction 𝐹 , abelfunction 𝐺 and non-integer iterates for any growing real-holomorphic
transfer function 𝑇 . This Book describes the sequence of attempts tonegate, refute this claim. All these attempts failed: I could not find any
transfer function, for which I cannot construct the superfunction. In
support of my claim, the Book presents examples of transfer function
with real fixed point(s), examples with complex fixed points, and theexample of the transfer function without any fixed point. While all the
tests are successful, the ability of construction of superfunctions can beinterpreted as a scientific fact.
I just have mentioned, what is done. But, as usually, a lot of can bedone about superfunctions. The next section is dedicated to this.

2

Future work

I tried to collect that I know about superfunctions and iterates, in thisBook. However, always some fantasies remain, what else would be interesting to do. Some of these fantasies are collected below.

293

The more efficient, more simple and more rigorous proofs of the existence
and uniqueness of superfunction may require the future work, and also
the additional conditions, that should be applied in order to provide theuniqueness of superfunction.
Application of superfunctions into the laser science may be subject for
the future analysis. Especially, this refers to the laser science, where thephysical sense of superfunctions and the non-integer iterates is especially
explicit.
The future work may be related to the specific case of superfunctions,
namely, ackermanns. Figure 19.7 shows the graphics for the first fiveackermanns: addition of constant, multiplication to constant, exponent,
tetration and pentation. I think, similar graphics can be constructedalso for higher ackermanns.
The important suggestion for the future work is related with the automatisation of construction of superfunctions. I mean the automaticalgorithm, that begins with the transfer function, searches for the appropriate fixed points of this function (if the transfer function has fixed
points), chooses the appropriate asymptotic for the superfunction, use
it to build-up the superfunction, build-up the corresponding abelfunction and calculates the non-initeger iterates. The software Mathematica
already has name for such a procedure; it is called Nest. Up to date of
preparation of this Book, the routine Nest can deal only with very special case, namely, when the number of iterate is expressed with positive
integer constant; in other cases, the call is interpreted as error. The upgrade of that routine for the real and complex number of iterate would
be interesting.
Fantasies and curiosity should be motivations for any serious scientificresearch. I collect the tools that can be used in this work. Many of them
are described in this Book.

3

Notations

I try to use the same notations through the whole Book. In order to approach this, some notations are different from those used in the original
publications. The notations are collected in tables 21.1, 21.2. I collectthe most important notations, and those, that often cause confusions.
294

Table 21.1: Notations, Latin alphabet(︁)︁𝒜(𝑚, 𝑧) = 𝒜 𝑚−1, 𝒜(𝑚, 𝑧−1)
𝐴𝑏,𝑛𝑎𝑛
AuExp𝑏 = SuExp−1𝑏canonical ackermann (19.14)
ackermann to base 𝑏 of order 𝑛, (19.7)
Coefficients in various asymptotic expansions
growing abelexponent to base 𝑏

AuTra

Abel function for the Trappmann function

AuSin

abelsineArc
prefix indicating the inverse function𝑏base of ackermanns, (19.1)
𝑐𝑛
coefficients in expansions, for ex., (10.47) or (20.15)
𝐶𝑛coefficients in expansions, for ex., (7.12)
e = exp(1) ≈ 2.7
base of natural logarithms
exp
exponent to base e
exp𝑏exponent to base 𝑏
𝐹
Name, used to denote superfunctions
Factorial(𝑧) = 𝑧!
holomorphic solution of equations 𝑧! = 𝑧 (𝑧 −1)!
𝑓name used for various functions−1
𝐺=𝐹
name used to denote abelfundtions
0.5ℎ = Elu𝑠
half iterate of logistic operator√
2
imaginary unit i = −1
i = −1𝑖variable, usually it takes integer values𝑘𝐾
plug for the contour integral rfK, or e𝑘increment in the asymptotic representation of superfunction (6.3)
𝐿
fixed point of transfer function 𝑇 , id est, solution of𝑇 (𝐿) = 𝐿lnlogarithm to base elog𝑏Logarithm to base b𝑀number of terms in truncated series𝑂some function that grows-up not faster than its argument
𝑃
period or asymptotic period; also is used as inverse of function 𝑄
𝑄
is used as inverse function of 𝑃 in table 3.1
Super
prefix to name of function, indicating its sperfunction𝑠parameter of the logistic operator (7.1)
𝑇transfer function
tetnatural tetration (14.28), [54]
𝑓 (𝑧)
tet𝑏
solution 𝑓 of equaitons 𝑓 (𝑧 +1) = 𝑏, 𝑓 (0) = 1, see fig. 17.1
𝑢real part of function in the complex map𝑣
imaginary part of function in the complex map
𝑥used as dummy parameter, that takes real values
𝑦used as dummy parameter, that takes real values𝑧used as dummy parameter, that takes complex values; often 𝑧 = 𝑥+i𝑦
295

Table 21.2: Notations, Greek alphabet

𝛼,𝛽,𝛾used as parameters in the expansions
𝜂 = exp(1/e) ≈ 1.444667861
Henryk base𝜀 = exp(𝑘𝑧)small parameter in the expansion of superfunctions
𝜃holomorphic periodic function with period unity∞∑︁1, Riemann function (8.9), [13, 18]𝜁𝜁(𝑧) =𝑛𝑧
𝑛=1
Ω4

contour of integration in the Cauchy formula (14.21)

Afterwards

The first (Russian) version of this Book happened thicker, than I had
expected. It could be much thicker; because all the time, there is theevil illusion, that some additional small formula should greatly simplify
the understanding. However, with each formula from the book, several
additional pictures can be plotted. However, I think, for the Reader,
it will be much better, to plot some picture, than to see the gallery ofsimilar figures in the Book.
I tried to make this Book shorter, than its first Russian version. Actually,
the Book happened to be longer, thicker, because I include here thechapter about the Nemtsov function Nem𝑞 (𝑥) = 𝑥 + 𝑥3 + 𝑞𝑥4 ; this isimportant example of the exotic iteration of transfer function with thespecific expansion at the fixed point.
Some things are still dropped out from the Book. I did not include
the holomorphic extension of tetration beyond the cut lines. I did notinclude figures to iterates of the exponent to complex base. And I did notinclude many other figures, assuming, that the Readers can download
the generators of the figures and plot all modifications they need.
The Readers are invited to download algorithms, figures and their generators from https://mizugadro.mydns.jp/t/index.php/Category:
Book; with these tools, the colleague may continue from that points ofresearch, where I am now. I believe, this is correct style of making
science, where all the results are available for researchers. This is supposed to simplify the verification and refutation (if anything is seriously
wrong), as well as revealing and correction of mistakes in formulas andbugs in the algorithms. I invite the colleagues to expose their results ina similar way.
296

Fig. 21.2: Near relatives, whose existence is essential for writing of this Book

5

Acknowledgement

I am grateful to relatives (Figure 21.2) for the tolerance with respect tothis Book, it took much more time and efforts, than expected.
I am grateful to colleagues, who helped
me to collect the literature on superfunctions (Figure 21.3) and arrange theserver, database and mediawiki: without these instruments, I would get lost
among a thousand files used for generation of this Book.
The goal was to get possibility to answer
the questions on superfunctions with eiFig. 21.3: Bookorm [119]ther Nobody knows this! or Das ist inmeinem Buch! (see figure 21.4). Perhaps, this goal is not reachable; the new algorithms are expected to bereported [94]. Tanks to colleagues who keep doing superfunctions.

Fig. 21.4: Das ist in meinem Buch!

297

Chapter 22
Supplement
1

About the coverAt the front cover of this book, the map is shown, from https://

mizugadro.mydns.jp/t/index.php/File:Tetma.jpg
This is one of versions of figure 14.12; Some details and labels are removed in order to simplify the aesthetic view.
The cover of the Russian version is loaded as https://mizugadro.

mydns.jp/t/index.php/File:Covervi.jpg

298

Fig. 22.1: The reader should have Maple-10 installed, in order to reproduce the plots
above with the codes supplied at the top of each picture2

Maple and tea

Here is the company C of computers; they master:
Per each two years, their PCs run twice faster.
Here is company S of the soft; they work hard as plowers,
So, each new release runs 0.7 times slower;
The soft is to run at the user’s PC,
Which is made, of course, by the company C.
Here is user U, buys from C and from S,
the newest model and the latest release.
U presses some key, say, key number K;
The soft S responds, with some delay,
during some time, say, during time t,
to let user U to have some tea.
How many releases per year does sell S to U
to let him have tea, while he has nothing to do,just waiting response by the soft to key K,
keeping the same time t of delay,
neutralizing the efforts of company C
to run faster U’s task at his modern PC?
This problem above is not correct, because
The tea-concern, together with C, of course,
support the efforts of the company S
to make even bigger the newest release,To force U buy more tea and a newest PC,
to boost business of C and the concern of tea.
(2007)299

https://mizugadro.mydns.jp/t/index.php/File:Ausintay40t50.jpg

−1
=
𝑢

𝑢=

1

−2

𝑦

𝑢=

−1

0

𝑣=

1

0

𝑣 =0

𝑣 = 0.2𝑣 =0𝑣 = −0.2

𝑢=

𝑣 =1

𝑣 =3
𝑣 =2

0

𝑢=2
𝑢=1

𝑢 = −0.5

−1

0

𝜋/2

2

3𝑥

Fig. 22.2: Map of the Taylor expansion 𝑓 of AuSin with 40 terms in the standard
notations. 𝑢+i𝑣 = 𝑓 (𝑥+i𝑦).

3

Taylor expansion of AuSin

In chapter 13, the Taylor expansion of function AuSin at 𝜋/2 is mentioned, (12.24). In figure 22.2, the complex map of the truncated series
with 40 terms is shown. This map should be compared to Figure 12.4.
Such an expansion can be used to boost the precise evaluation of AuSin
for values of the argument in vicinity of 𝜋/2. Also, coefficients of this expansion can be used for evaluation of coefficients of expansion of SuSun
at zero; for example, wight routine InverseSeries.
300

4

Sites

Goal to his Book is not to attract attention of Reader to the cites below,
but to provide the description of superfunctions, that releases the reader
from need to browse the sites below.
https://allmybase.com/dropbox/tetration.pdf
https://en.citizendium.org/wiki/Cauchy-Riemann_equations
https://en.citizendium.org/wiki/Superfunction
https://en.citizendium.org/wiki/Tetrational
https://en.citizendium.org/wiki/Holomorphic_function
https://en.wikipedia.org/wiki/Abel_equation
https://en.wikipedia.org/wiki/Abel_function
https://en.wikipedia.org/wiki/Cauchy’s_integral_formula
https://en.wikipedia.org/wiki/Hellmuth_Kneser
https://en.wikipedia.org/wiki/Niels_Henrik_Abel
https://en.wikipedia.org/wiki/Superfunction
https://en.wikipedia.org/wiki/Tetration
https://math.eretrandre.org/tetrationforum/index.php
https://math.stackexchange.com/tags/tetration
https://mathworld.wolfram.com/Tetration.html
https://oeis.org/wiki/Tetration
https://mizugadro.mydns.jp/t/index.php/Abel_function
https://mizugadro.mydns.jp/t/index.php/ArcShoka
https://mizugadro.mydns.jp/t/index.php/AuSin
https://mizugadro.mydns.jp/t/index.php/AuTra
https://mizugadro.mydns.jp/t/index.php/AuZex
https://mizugadro.mydns.jp/t/index.php/Complex_map
https://mizugadro.mydns.jp/t/index.php/Doya_function
https://mizugadro.mydns.jp/t/index.php/Factorial
https://mizugadro.mydns.jp/t/index.php/Holopmorphic_extension_of_Collatz_
Subsequence
https://mizugadro.mydns.jp/t/index.php/Keller_function
https://mizugadro.mydns.jp/t/index.php/LambertW
https://mizugadro.mydns.jp/t/index.php/Logistic_sequence
https://mizugadro.mydns.jp/t/index.php/Regular_iteration
https://mizugadro.mydns.jp/t/index.php/Shoka_function
https://mizugadro.mydns.jp/t/index.php/Superfunction
https://mizugadro.mydns.jp/t/index.php/Superfactorial
https://mizugadro.mydns.jp/t/index.php/SuSin
https://mizugadro.mydns.jp/t/index.php/SuTra
https://mizugadro.mydns.jp/t/index.php/SuTra
https://mizugadro.mydns.jp/t/index.php/Table_of_superfunctions
https://mizugadro.mydns.jp/t/index.php/Tania_function
https://mizugadro.mydns.jp/t/index.php/Tetration
https://www.proofwiki.org/wiki/Definition:Superfunction
https://www.proofwiki.org/wiki/Definition:Tetration
https://www.tetration.org/Tetration/index.html D.Geisler. What lies beyond exponentiation?

https://www.youtube.com/watch?v=z-mfxP1Tmfw Kasane Teto. Tetration ↑↑. 2012.

301

5

New notations

In future, the new notations will be requested. Below, I suggest tentative
names for them:Transferation. For given function 𝑓 , construction of the transfer func-tion 𝑇 (𝑧) = 𝑓 1+𝑓 −1 (𝑧) . For this transfer function 𝑇 , function 𝑓 issuperfunction and 𝑓 −1 is abelfunction.

(︀)︀Transation. For given function 𝑓 , construction of the transfer function

n

io

ct

n

u
F
se

r
ve
In

−→

← transferation ←−→ superation −→

−
←

(︀
)︀
𝑇 (𝑧) = 𝑓 −1 1 + 𝑓 (𝑧) . For this transfer function 𝑇 , function 𝑓 −1 issuperfunction and 𝑓 is abelfunction.
Superation. For given function 𝑓 , construction of superfunction 𝐹 as(︀)︀solution of equation 𝐹 (𝑧 + 1) = 𝑓 𝐹 (𝑧) . In this case, 𝑓 appears astransfer function.
Supation. For given function 𝑓 , construction of abelfunction 𝐺 as(︀)︀
solution of the Abel equation 𝐺 𝑓 (𝑧) = 𝐺(𝑧)+1. In this case also 𝑓
appears as a transfer function.
Transferation is inverse operation with𝐹
respect to supperation. Transfation isinverse operation with respect to supation. This is shown in figure 22.3 forthe transfer function 𝑇 , superfunction
𝐹 and abelfunction 𝐺.

−→ supation −→𝑇 ← transfation ← 𝐺This Book is dedicated to superationand supation, and also to the additional
requirements, that provide the uniqueness of superfunction 𝐹 and abelfunction 𝐺.

In this Book, I do not use the new words
𝑇 , 𝐹 and 𝐺 = 𝐹 −1
shown in figure 22.3; while, there is no need to use them. In the similarway, there was no need in the special terms for differentiation and integration, until these operations became routines. However, the terms
shown in figure 22.3 will be requested as soon as the automatic construction of superfunctions and abelfunctions will be realised. In language
Mathematica, for the superation, there exist name Nest (until year 2017,
this routine is supported only for natural values of the number of iterate).
Fig. 22.3:

I am not sure, which names from figure 22.3 will be usual. For thisreason, I do not use these names in the Book. However, the operations,
mentioned in figure 22.3 exist; they will require some names.
302

6

Extended abstract

The Book is dedicated to construction of superfunction 𝐹
and abelfunction 𝐺 for given transfer function 𝑇 . The formalism of superfunctions is considered with examples, thatare presented in ready-to-use form. The reader is supposed toknow something about the complex numbers and elementaryfunctions.
The superfunction
is solution of the transfer equation(︀)︀𝐹 (𝑧 +1) = 𝑇 𝐹 (𝑧)Theabelfunction
𝐺 = 𝐹 −1 satisfies the Abel equation(︀)︀
𝐺 𝑇 (𝑧) = 𝐺(𝑧) + 1
In order to provide the uniqueness of solution, the additional
requirements on 𝐹 are applied, referring to its behaviour inthe complex plane.
The 𝑛th iterate of function 𝑇 is denoted with superscript:
(︁ (︀
)︀)︁
𝑇 𝑇 ..𝑇 (𝑧)..
𝑇 𝑛(𝑧) =⏟⏞𝑛 evaluations of function 𝑇
With superfunction 𝐹 andabelfunction
𝐺, the iterate is ex(︀)︀pressed as 𝑇 𝑛(𝑧) = 𝐹 𝑛+𝐺(𝑧) . In this representation,
the number 𝑛 of iterate has no need to be integer.
Examples of superfunctions are considered and collected as
Table 3.1. Superfunctions are constructed for sin, factorial,
exponential, tetration and other functions.
Many explicit plots and complex maps for these functions,superfunctions, and iterates are included. The figures areloaded also to TORI together with their generators at
https://mizugadro.mydns.jp/t/index.php/Category:Book

The formalism of superfunctions greatly extends the set offunctions available for applications in the scientific research.
303

7

Keywords

In this section, I suggest the essence of the notations

Transfer function𝑇

𝑇 𝐹 (𝑧) = 𝐹 (𝑧 +1)
Transfer equation, superfunction(︀)︀
𝐺 𝑇 (𝑧) = 𝐺(𝑧) + 1
Abel equation, abelfunction(︀)︀𝐹 𝐺(𝑧) = 𝑧
Identity function(︀)︀𝑛
𝑇 (𝑧) = 𝐹 𝑛 + 𝐺(𝑧)𝑛th iterate∮︁1
𝐹 (𝑡) d𝑡
𝐹 (𝑧) =
Cauchy integral
2𝜋i
𝑡−𝑧
(︀

)︀

tetration to base 𝑏(︀)︀tet𝑏 ate𝑏(𝑧) = 𝑧tet𝑏(𝑧 +1) = 𝑏tet𝑏(𝑧)tet𝑏(0) = 1 ,ate𝑏(𝑏𝑧 ) = ate𝑏(𝑧) + 1
arctetration to base 𝑏(︀)︀𝑛exp𝑏 (𝑧) = tet𝑏 𝑛+ate𝑏(𝑧)𝑛th iterate of function 𝑧 ↦→ 𝑏𝑧
Tania(𝑧)Tania function, Tania(0) = 1
Tania(𝑧)+1(︀)︀
Doya(𝑧) = Tania 1+ArcTania(𝑧)Doya function

Tania′(𝑧) =Shoka(𝑧) = 𝑧 + ln(e−𝑧 +e−1) Shoka function(︀)︀Keller(𝑧) = Shoka 1+ArcShoka(𝑧)Keller function
tra(𝑧) = 𝑧 + exp(𝑧)Trappmann function

zex(𝑧) = 𝑧 exp(𝑧)Zex function

304

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D.Kouznetsov, E.Gomez. Self-Modulation of Optical Pulses
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arnumber=1278604
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D.Kouznetsov, J.V.Moloney. Slab delivery ofincoherent pump light to double-clad fibre amplifiers: An analytic
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В.Садовничий. ПЕРВОМУ УНИВЕРСИТЕТУ
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pdf
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D.Kouznetsov, J.-F.Bisson, A.Shirakawa, K.Ueda. Limits ofcoherent addition of lasers: simple estimate. – Optical Review,
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D.Kouznetsov, J.-F.Bisson, K.Takaichi, K.Ueda. High-power
single mode solid state laser with short unstable cavity. JOSA B,
v.22, Issue 8, p.1605-1619 (2005).
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p221117_s1 Wenxue Li, Haifeng Pan, Liang’en Ding, Heping Zeng, Wei Lu, Guangjun Zhao, Chengfeng Yan, Liangbi
Su, and Jun Xu. Efficient diode-pumped Yb : Gd2 SiO5 laser.
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Chunlei Guo, Guangjun Zhao, Chengfeng Yan, Liangbi Su,
and Jun Xu. Efficient tunable diode-pumped Yb:LYSO laser.
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S0038109805010094
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C.Yan, G.Zhao, L.Zhang, J.Xu, X.Liang, D.Juan, W.Li, H.Pan,
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90_066101.pdf
https://mizugadro.mydns.jp/PAPERS/ApplPhysLett_90_
066101.pdf D. Kouznetsov. Comment on Efficient diode-pumped
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[46] https://mizugadro.mydns.jp/PAPERS/2007mc.pdf
D.Kouznetsov. Broadband laser materials and the McCumber relation. Chinese Optics Letters v.5, p.S240–S242 (2007)[47] https://apl.aip.org/resource/1/applab/v90/i6/p066103_
s1 G.Zhao. Response to Comment on Efficient diode-pumped
Yb:Gd2SiO5 laser. Appl. Phys. Lett. v.90, 066101 (2007)[48] https://mizugadro.mydns.jp/PAPERS/ApplPhysLett_90_201901.pdf
J.-F.Bisson, D.Kouznetsov, K.Ueda, S.T.Fredrich-Thornton,
K.Petermann, G.Huber. Switching of emissivity and photoconductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics. –Applied Physics Letters, 90, 201901 (2007) ; doi:10.1063/1.2739318
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J.-F.Bisson, D.Kouznetsov, K.Ueda, T.Fredrich-Thornton,
K.Petermann, G.Huber. Switching of emissivity and photoconductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics. –Applied Physics Letters, 90, 201901 (2007).
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D.Kouznetsov. Maple and Tea. (2007)[52] https://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf
В.М.Гордиенко, В.К.Новик. О времени и факультете, окафедре и о себе... 70-ти летие профессора В.П.Кандидова.
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Acta Physica Polonica B, 2008, v.39, p.1019.
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pdf

D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex
z-plane. Mathematics of Computation 78 (2009), 1647-1670.
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S1068335609050054#page-1 Gorbuklov M.V., Maslova Yu.Ya.,
Chaban O.I., Shabalin Yu.V. Period doubling cascade and deterministic chaos in a laser self-mode-locked by the combination ofintertial negative and positive feedbacks. Bulletin of the Lebedev
Physics Institute, 36. No.5. (2009). P.150-156.
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d73u677140l3j826/

Bjerklöv K. SNA’s in the Quasi-Periodic Quadratic Family
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[57] https://mizugadro.mydns.jp/t/index.php/Lesson_of_quantum_topology[58] P.V.Elutin. The holomorphic extension of the logistic mapping isambitious program... The continual generalization of the logistic
mapping will worth your time and efforts, if you can reveal themodel that gives the law of the evolution 𝐹 (𝑡) - or determine thecharacteristics 𝐹 (𝑡), boundaries of the time derivatives. - Private
communication, Oct 9, 2009 5:32 PM. (Елютин П. В. Думаю,задача,понятая как “КОНТИНУАЛЬНОЕ ОБОБЩЕНИЕ

ЛОГИСТИЧЕСКОГО

ОТОБРАЖЕНИЯ”,будетстоить

Ваших усилий и времени, если удастся выявить модель,порождением которой является закон движения 𝑋(𝑡) - илиопределить континуальные характеристики 𝑋(𝑡) (например,границы значений производных по времени)[59] https://iopscience.iop.org/1742-6596/237/1/012016/

pdf/1742-6596_237_1_012016.pdf

P.G.L.Leach and M.C.Nucci. Point and counterpoint between
Mathematical Physics and Physical Mathematics. Journal of
Physics: Conference Series 237 (2010) 012016.
[60] https://projecteuclid.org/euclid.jpm The Journal of Physical Mathematics is a scholarly, peer-reviewed, and fully refereed
journal dedicated to publishing high-quality research articles andselected review articles in the fields of physically motivated mathematics and mathematical methods of physics. // The Journal of
Physical Mathematics was published by Ashdin Publishing until
2013. Beginning in 2017, this journal will no longer be included in
Euclid Prime or distributed by Project Euclid.
[61] https://www.ams.org/journals/mcom/2010-79-271/

S0025-5718-10-02342-2/home.html
http://eretrandre.org/rb/files/Kouznetsov2009_215.pdf
https://www.ils.uec.ac.jp/~dima/PAPERS/2010q2.pdf
https://mizugadro.mydns.jp/PAPERS/2010q2.pdf

D.Kouznetsov, H.Trappmann. Portrait of the four regular superexponentials to base sqrt(2). Mathematics of Computation, 2010,
v.79, p.1727-1756.
[62] https://www.springerlink.com/content/

qt31671237421111/fulltext.pdf?page=1
https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf

D.Kouznetsov, H.Trappmann. Superfunctions and square root of312

factorial. Moscow University Physics Bulletin, 2010, v.65, No.1,
p.6-12[63] http://www.vmj.ru/archive/detail.php?ELEMENT_ID=3145&

SECTION_ID=337 http://www.vmj.ru/articles/2010_2_4.pdf
https://mizugadro.mydns.jp/PAPERS/2009vladir.pdf
https://www.ils.uec.ac.jp/~dima/PAPERS/2009vladir.pdf
Д.Кузнецов.
Тетрация
какспециальная
функция.
Владикавказский математический журнал, 2010, т.12, вып. 2,
стр.31-45. (In Russian. See the English version below.)[64] https://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf

https://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
D.Kouznetsov. Tetration as special function.
Mathematical Journal, 2010, v.12, issue 2, p.31-45.

Vladikavkaz[65] https://cyberleninka.ru/article/n/superfunktsii-i-koren-iz-faktoriala/viewer
https://mizugadro.mydns.jp/PAPERS/2010superfar.pdf

Кузнецов Д.Ю.Траппманн Г. Суперфункции и корень изфакториала. Вестник Московского Университета, серия 3,2010 No.1, стр.8-14. (Текст научной статьи по специальности
«Математика»), in Russian. See the English version below:[66] https://link.springer.com/article/10.3103/

S0027134910010029
http://www.ils.uec.ac.jp/~dima/PAPERS/2010superfae.
pdf https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf
https://www.researchgate.net/publication/226675861_
Superfunctions_and_sqrt_of_factorial
D.Kouznetsov,
H.Trappmann. Superfunctions and square root of factorial.
Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.

[67] https://pdfs.semanticscholar.org/fd21/

ac0be8deaa126a45849935bacd4cef2b7144.pdf H.Trappmann,
D.Kouznetsov. 5+ methods for real analytic tetration. June 28,
2010.

[68] https://ufn.ru/tribune/trib120111.pdf

https://mizugadro.mydns.jp/t/index.php/\T2A\CYRM\T2A\
cyre\T2A\cyrs\T2A\cyrt\T2A\cyro_\T2A\cyrn\T2A\cyra\
T2A\cyru\T2A\cyrk\T2A\cyri Д.Кузнецов. Место науки ифизики в человеческом знании. УФН, т.181 “Трибуна”, стр.1-9(2010, In Russian. See the English version below).
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https://mizugadro.mydns.jp/t/index.php/Place_of_science https://mizugadro.mydns.jp/t/index.php/Mestoe

D.Kouznetsov. Place of science in human knowledge. New Insights
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https://mizugadro.mydns.jp/PAPERS/2010logistie.pdf
https://link.springer.com/article/10.3103/
S0027134910020049 D.Kouznetsov. Holomorphic extensionof the Logistic sequence. Moscow State University Physics Bulletin, v.65, p.91-98 (2010). The logistic problem is formulated interms of the Superfunction and Abelfunction of the quadratictransfer function H(z)=uz(1-z). The Superfunction F as holomorphic solution of equation H(F(z))=F(z+1) generalizes thelogistic sequence to the complex values of the argument z. The
efficient algorithm for the evaluation of function F and its inverse
function, id est, the Abelfunction G are suggested; F(G(z))=z.
The halfiteration h(z)=F(1/2+G(z)) is constructed; in wide rangeof values z, the relation h(h(z))=H(z) holds. For the special caseu=4, the Superfunction F and the Abelfunction G are expressedin terms of elementary functions.

[71] http://budclub.ru/k/kuznecow_d_j/mizugade.shtml

https://mizugadro.mydns.jp/t/index.php/Mizugadro_
number_dream D.Kouznetsov. Mizugadro’s number. (2010-2017)[72] Hitoki Yoneda. Recovery of intensity inside the amplifier from its
transfer function is impossible. Private communication, 2010.
[73] Aleksander Kaminskii. There is no sense to analyse the transfer
function of the amplifier without to specify the model for the active
medium. Private communication, 2011.
[74] https://link.springer.com/article/10.1007/

s00010-010-0021-6
https://pdfs.semanticscholar.org/
203a/d0586b625d2648aa6e22f36da7189f9041bd.pdf https://www.ils.uec.ac.jp/~dima/PAPERS/2011uniabel.pdf
https://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf

H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel
Functions in Absence of a Real Fixed Point. Aequationes
Mathematicae, v.81, p.65-76 (2011)314

[75] Akira Shirakawa. It is not possible to reconstruct the intensity
inside the amplified just from its transfer function. Private communication, 2011.
[76] Akira Shirakawa. The formalism of superfunctions is too complicated and therefore cannot be used. Private communication,2012.10.09.
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Kursernas Hemsidor. Introduction to Dynamical Systems. 2012.
Derivation of Niclas Carlsson’s formula. Let function 𝑓 (𝑥) be𝑠𝑖𝑛(𝑥). We want to evaluate, approximately .the value of the n-th
iterate of 𝑓 (𝑥). Niclas√︁ Carlsson obtained empirically the following
formula 𝑓 𝑛 (𝑥) ≈3𝑛 , 𝑥 ≈ 1, 𝑛 large .. If the formula is correct

it will take 3 ·1010 .. steps to reach 0.00001 starting from 1. ..After
60000 iterations the value is 0.0071 and the discrepancy -4.7323e007.
[78] http://www.physics.miami.edu/~curtright/Schroeder.
html Thomas Curtright. Continuous iterates continue to be interesting, after 150 years of study. (2011-2013) As a first illustration,
we display the continuous iterates of the sine function, sin[𝑡] (𝑥).
Note that the maximum
values at 𝑥 = 𝜋/2 are approximately√
given by exp[(1 − 𝑡) ln(𝜋/2)].
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Л.С.Сычева.
Физическая
математика
Архимеда,формирование интегрального исчисления и механизмы
новаций в математике. Школа, том 6, номер 2 (2012, in
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https://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.pdf
https://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf
H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics ofComputation, 2012, 81, p.2207-2227.

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Квашенко
Андрей Николаевич. Драконы. (2007-2013, in Russian)[82] https://mizugadro.mydns.jp/PAPERS/2013physToday.pdf
D.Kouznetsov. Corruption in Russian science. Preprint, 2013.
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Д.Кузнецов. Уничтожение науки в РФ. (Russian version)[84] https://www.scirp.org/journal/PaperInformation.aspx?PaperID=36560

https://mizugadro.mydns.jp/PAPERS/2013jmp.pdf

D.Kouznetsov. TORI axioms and the applications in physics.
Journal of Modern Physics, 2013, v.4, p.1151-1156.
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00000019/00000003/art00071
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D.Kouznetsov. Recovery of Properties of a Material from Transfer
Function of a Bulk Sample (Theory). Advanced Science Letters,
Volume 19, Number 3, March 2013, pp. 1035-1038(4).
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https://mizugadro.mydns.jp/PAPERS/2013orSuper.pdf

D.Kouznetsov. Superfunctions for amplifiers. Optical Review,
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Intuitive Logarithm. October 14, 2013.
[88] Michael Moldenhauer. Question about tetration method. Private
communication, Sat, 9 Nov 2013 01:11:53 -0800 (PST). ..how did
you come up with the high-quality initial approximation fit_3 mentioned in your paper about the tetrational function?

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https://pdfs.semanticscholar.org/6b95/
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Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.6527-6541.
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paperinfo.aspx?journalid=147&doi=10.11648/j.acm.
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phic Ackermanns. Applied and Computational Mathematics
Volume 3, Issue 6, December 2014, Pages: 307-314[91] https://dx.doi.org/10.1007/s00020-014-2175-4
Jose
Bonet, Pawel Domanski. Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic
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SamuelCowgill.
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https://www.ils.uec.ac.jp/~dima/e/Keller[111] https://mizugadro.mydns.jp/t/index.php/Shoka_function

https://www.ils.uec.ac.jp/~dima/e/Shoka[112] https://mizugadro.mydns.jp/t/index.php/Holomorphic_

extension_of_the_Collatz_subsequence[113] https://mizugadro.mydns.jp/t/index.php/LambertW[114] https://mathworld.wolfram.com/LambertW-Function.html[115] https://en.citizendium.org/wiki/Lambert_W_function[116] https://en.wikipedia.org/wiki/Lambert_W_function[117] https://www.researchgate.net/publication/228724354_
The_Wright_o_Function Robert M. Corless, David J. Jeffrey.
The Wright 𝜔 function. Artificial intelligence, January 2002.
[118] http://valerie.doya.free.fr/Doya_CV.html Valérie DOYA.
Laboratoire de Physique de la Matière Condensée, Parc Valrose,
06108 Nice Cedex2, France. (2014)[119] https://commons.wikimedia.org/wiki/File:Carl_
Spitzweg_021.jpg Carl Spitzweg. The Bookworm. 1850.

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Tools for evaluation of superfunctions, abelfunctions and non-integer
iterates of holomorphic functions are collected. For a given transfer
function T, the superfunction is solution F of the transfer equation
F(z+1)=T(F(z)) . The abelfunction is inverse of F. In particular,
superfunctions of factorial, exponent, sin are suggested. Also, theholomorphic extensions of the logistic sequence and those of the
Ackermann functions are considered. Among ackermanns, the tetration(mainly to the base b>1) and natural pentation (to base b=e) arepresented. The efficient algorithm for the evaluation of superfunctions
and abelfunctions are described. The graphics and complex maps
are plotted. The possible applications are discussed. Superfunctions
significantly extend the set of functions that can be used in scientificresearch and technical design. Generators of figures are loaded to thesite TORI, https://mizugadro.mydns.jp for the free downloading. With
these generators, the Readers can reproduce (and modify) the figuresfrom the Book. The Book is intended to be applied and popular.
I try to avoid the complicated formulas, but some basic knowledgeof the complex arithmetics, Cauchy integral and the principles of theasymptotical analysis should help at the reading.

Dmitrii Kouznetsov

Graduated from the Physics Department of the
Moscow State University, 1980. Works: USSR, Mexico, USA, Japan. Proven the quantum stability of the
optical soliton. Suggested the low bound of the quantum noise of nonlinear amplifier. Indicated the limitof the single mode approximation in the quantum optics. Theorem on the boundary behiaviour of modes ofDirichlet Laplacian. Theory of ridged atomic mirrors.
TORI axioms.