MATHEMATICS OF COMPUTATION
Volume 00, Number 0, Pages 000–000
S 0025-5718(XX)0000-0

PORTRAIT OF THE FOUR REGULAR SUPER-EXPONENTIALSTO BASE SQRT(2)DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

Abstract. We introduce the concept of regular super-functions at a fixed
point. It is derived from the concept of regular iteration. A super-function
F of h is a solution of F(z+1)=h(F(z)). We provide a condition for F being
entire, we also give two uniqueness criteria for regular super-functions.
In the particular case h(x)=bˆx we call F super-exponential. h has tworeal fixed points for b between 1 and eˆ(1/e). Exemplary we choose the baseb=sqrt(2) and portrait the four classes of real regular super-exponentials in thecomplex plane. There are two at fixed point 2 and two at fixed point 4. Eachclass is given by the translations along the x-axis of a suitable representative.
Both super-exponentials at fixed point 4 — one strictly increasing and one
strictly decreasing — are entire. Both super-exponentials at fixed point 2 —one strictly increasing and one strictly decreasing — are holomorphic on aright halfplane. All four super-exponentials are periodic along the imaginary
axis. Only the strictly increasing super-exponential at 2 can satisfy F(0)=1
and can hence be called tetrational.
We develop numerical algorithms for the precise evaluation of these functions and their inverses in the complex plane. We graph the two corresponding
different half-iterates of h(z)=sqrt(2)ˆz. An apparent symmetry of the tetrational to base sqrt(2) disproved.

1. Super-Functions
The functional equation(1.1)

F (z+1) = h(F (z))

refers to the building-up of a super-function F of some given base-function h. Multiplication F (x) = bx is a super-function of addition h(x) = b + x. Exponentiation
F (x) = bx is a super-function of multiplication h(x) = bx. Such an equation also
occurs in the phenomenological description of a nonlinear optical fiber of fixed
length [11], where F (z) represents some parameter (for example, the logarithm of
the power) at distance z from the input tip, and h is the transfer function. In thatapplication z is supposed to be real. In general, we assume, that equation (1.1)
holds for z ∈ C ⊆ C, and F is holomorphic on C.
Definition 1 (super-function, base-function, super-exponential, tetrational). Let
C be a domain such that for each z ∈ C also z+1 ∈ C. A function F defined on C
is called super-function of h, if it satisfies equation (1.1) for all z ∈ C. We call h thebase-function of F . Applying the prefix a 7→ d (to “super-function”) means that
Received by the editor August 27, 2009.
2000 Mathematics Subject Classification.

Primary 30A99; Secondary 33F99 .
c XXXX American Mathematical Society

1

2

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

additionally F (a) = d. Sometimes we also include the domain of holomorphism C
into the prefix, writing (C, a 7→ d), which additionally means that F is holomorphic
on C. Any super-function of h(z) = bz is called super-exponential to base b. Any
0 7→ 1 super-exponential to base b is called tetrational to base b.
The name super-exponential is due to Bromer [6], although other names (generalized exponential, iterated exponential, tetration, ultraexponential, etc.) are also
used for the iteration of an exponential function [10, 9, 3, 16].
In order to illustrate the definition of super-function, we suggest few examples
below.
Let h(z) = b+z on z ∈ C. Then, the function F (z) = b ·z on z ∈ C is a (C, 0 7→ 0)
super-function of h.
Now let h(z) = b · z on z ∈ C. Then F (z) = expb (z) = bz is a (C, 0 7→ 1)
super-function of h.
Now we let the previous super-function again be the base-function h = expb . For
C = C\{x ∈ R : x ≤ −2}, the tetrational F by [11] is a (C, 0 7→ 1) super-functionof exp or super-exponential to base e .
For some base-functions it is possible to obtain elementary super-functions byusing special functional relations, for example F (z) = cos(π2z ) is a (C, 0 7→ −1)super-function of the quadratic base-function h(z) = 2z 2 − 1. More trigonometricexamples can be found in [18]. Also, for real j > 0, and base-function h(z) = z j
the function F (z) = exp(j z ) is a super-function.
No representation through elementary or special functions is reported for nonconstant super-exponentials yet. So it would make sense to lift them into thestatus of new special functions after clarifying the uniqueness situation (similar tothe complex extension of the factorial through the Gamma function).
Without additional conditions a super-function is not unique, generally. If F is a
(C, a 7→ d) super-function of h then for any 1-periodic
holomorphic function θ, suchthat θ(a) = 0, the function E(z) = F z +θ(z) will be a (D, a 7→ d) super-functionof h for D = {z : z+θ(z) ∈ C}:
(1.2)

E(a)

(1.3)

E(z+1)= F (a + θ(a)) = d= F (z + θ(z+1) + 1) = h(F (z + θ(z))) = h(E(z)) .

However, the range D of holomorphism of the function E may be different from C.
Here is a question how an appropriately big domain of holomorphism restricts thepossible solutions. We give a uniqueness criterion in proposition 7.
This paper is motivated by discussions on [17] about a generalization of thealgorithm [11] for a holomorphic tetrational, which seems to work for various valuesb > exp(1/e), in particular, for b = 2, see [12, 19], but cannot be applied “as is”
for the case 1 < b ≤ exp(1/e). In order to elaborate a robust algorithm√for theevaluation of the tetrational in this case, we consider one specific value b = 2 fromthis interval. We consider not only the tetrational, but also some super-exponentials
to this base which can be easily generalized to bases 1 < b < exp(1/e).
In order to be explicit, we look for holomorphic, real solutions F of the equation(1.4)
F (z+1) = expb F (z)√for the specific value b = 2. (We call any C 7→ C function f real, if f (z ∗ ) =f (z)∗ ∀z ∈ C.)

SUPER-EXPONENTIALS TO BASE

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3

2. Super-functions by regular iteration
In this section we assume, that the function h has some fixed point λ, id est
h(λ) = λ. For example λ = 2 and λ = 4 are the only real fixed points of h = exp√2and λ = e is the only real fixed point of h = expexp(1/e) . Assume that the function
h is holomorphic at least in some vicinity of the fixed point λ. Then, a superfunction F of h can be constructed (under certain additional conditions), such that
F approaches λ asymptotically, at least in some directions in the complex plane.
The holomorphic function h can be expanded in a series that converges in somedomain around the fixed point:(2.1)

f (z) = h(λ + z) − λ =∞X

fn z nn=1

In this paper, for a function f we use notation fn for the nth Taylor coefficientat λ. The formal solution F with exponential behavior can be constructed in thefollowing way. Let(2.2)

F (z) = λ + G(E(z)),

for some function G expandable into a series(2.3)

G(z) = z +∞X

Gn z n

n=2

and
(2.4)


E(z) = exp (z+t) κ = s exp(κz)

for some constant parameters t and κ, where s = eκt . Substitution z 7→ z +1 inequation (2.2) gives(2.5)

F (z+1) = λ + G(KE(z))

where K = exp(κ). Then, equation (1.1) can be written as
(2.6)
λ + G(KE(z)) = h λ + G(E(z))
(2.7)
λ + G(Kω) = h λ + G(ω)
The Taylor expansion of both sides of (2.7) at ω = 0 leads to the equalities for thecoefficients:
(2.8)

K

(2.9)

G2 K 2(2.10)(2.11)= f1= f1 G2 + f2

G3 K

3

= f1 G3 + 2f2 G2 + f3

G4 K

4

= f1 G4 + 3f3 G2 + 2f2 G3 + f2 G22 + f4

It seems as if Gn can be determined without solving polynomial equations. This isindeed the case and it turns out that this approach is equivalent to the well-known
approach of regular iteration via the Schröder equation [15, 14, 7].
Few notes about notations.
For any (formal) powerseries f we denote the coeffiP∞
cients by fn ; f (z) = n=0 fn z n (although the series has no need to converge.) We

4

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

write f m for the mth power of the function f , and f [m] for the mth iterate of f , idestf [m] (z) = f f ...f (z)...
(2.12)|{z}m evaluations of f
For example, below f m n means the nth Taylor coefficient of the function that is
the mth power of function f . It should not be confused with fn m , which is the mth
power of the nth Taylor coefficient, or with f [m] n , which is the nth coefficient of the
expansion of the mth iterate of function f at 0. There are few exceptions from thisrule. First, we keep the usual notation F −1 for the inverse function of F , although
it is actually minus first iteration of function F ; and of course we use subscripts
as usual on non-functions, like κ2 or T4 . You can find a table of notations in the
appendix; we tried to distribute the semantic load uniformly among the letters of
Latin alphabet, but use few Greek letters too.
Definition 2 (Schröder equation, Schröder function). For a function h with fixed
point λ and K = h0 (λ), we call any function χ that satisfies the Schröder equation(2.13)

χ(h(z)) = Kχ(z)a Schröder function of h at λ.
For ease of derivation we first assume that the fixed point is 0. If we want the
Schröder function χ to be analytic, then it is already determined by χ0 (0) = 1. Weshow this in the next proposition.
Proposition 1 (Schröder powerseries). Let f be a formal powerseries with f0 = 0(fixed point 0) and f1 n 6= f1 for all integer n > 1 (particularly if |f1 | 6= 0, 1).
Then there is a unique formal powerseries solution χ with χ1 = 1 of the Schröderequation(2.14)

χ(f (z)) = f1 χ(z).

We call this solution the Schröder powerseries of f . The Schröder powerseries χand its inverse η = χ−1 are given by χ0 = η0 = 0, χ1 = η1 = 1, and, for n > 1, by therecursive formulas(2.15)

χn =

n−1
X
1
χm f m n ,n
f1 − f1 m=1

ηn =

n
X
1
fm η m n
f1 − f1 m=2nwhere the mth power of an arbitrary formal powerseries g is defined byX(2.16)
gm n =
gn1 . . . gnm .
n1 +···+nm =n

Proof. The recursive solutions are a direct consequence of the composition formulaof two formal powerseries f and g with g0 = 0:nX
(2.17)(f ◦ g)n =fm g m nm=0applied to our Schröder equation (2.14) or its inverse η(f1 z) = f (η(z)):
nnXX
f1 χn =
χm f m n ,f 1 n ηn =fm η m n .
m=0

m=1

SUPER-EXPONENTIALS TO BASE

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First we can observe at the left side that f1 χ0 = χ0 which implies χ0 = 0 for
f1 6= 0, 1. In consequence η0 = 0. For the indices above 1 we rearrange theequations to get the χn or ηn that occur in the sum of the right side to the left side
and where we take into account that f n n = f1 n and η 1 n = ηn .
Proposition 2. Every formal powerseries X solution of (2.14) can be expressed
as X (z) = sχ(z) for some s ∈ C.
Proof. Setting X1 = s in the above derivation, one inductively gets Xn = sχn .



Now we apply this to our original problem (2.7). We see that G is just the inverse
Schröder function of h(z+λ) − λ. Let us write down the first few coefficients of G.
G0 = 0 ,(2.18)

G2 =f2
1
f2 g 2 2 = 2
f12 − f1
f1 − f13
f2 (2G1 G2 ) + f3
n1 +n2 =3 Gn1 Gn2 + f3 G3=
G3 =3
f1 − f1
f13 − f1
2
2f2 f 2f−f
+ f3
2f2 f2 + f3 f12 − f3 f1
2f 2 + f3 f12 − f3 f1
1
1
=
=
= 2
3
3
2
f1 − f1(f1 − f1 )(f1 − f1 )(f1 −1)(f12 −1)f12

f2(2.19)

G1 = 1

P

Proposition 3. If the powerseries f with f0 = 0 and |f1 | =6 0, 1 has non-zeroconvergence radius, then the Schröder powerseries χ of f has non-zero convergenceradius too.
Proof. For a proof you can see [14], theorem 4.6.1.

Proposition 4. Let f be holomorphic in a vicinity of 0, f (0) = 0, set K = f 0 (0),let |K| 6= 0, 1. If f is entire and |K| > 1 or if f −1 is entire and |K| < 1 thenthe inverse Schröder function of f (i.e. the function with the inverse Schröderpowerseries development at 0) is entire.
Proof. We continue the inverse Schröder function to the whole complex plane. By
proposition 3 there is a disk Dδ with radius δ > 0 at 0 where the inverse Schröder±1function η of f is analytic. We then define η̂ on the disk with radius |K| δ for allz ∈ Dδ via(2.20)(2.21)

η̂(Kz) := f (η(z))η̂(z) := f−1

(η(Kz))

for|K| > 1

for|K| < 1η̂ coincides by the Schröder equation with η on Dδ and is hence a continuation to
D|K|±1 δ ⊃ Dδ . We repeatedly apply this process to continue η to the disk with±nradius |K| δ and so continue η to the whole complex plane.
Proposition 5. Let f be analytic at the fixed point 0, set K = f 0 (0) and assume|K| 6= 0, 1. Then, the Schröder powerseries χ of f can be analytically continued tothe basin of attraction Ω(f ) = {z : limn→∞ f [n] (z) = 0} of f at 0 (for |K| < 1) orto Ω(f −1 ) (for |K| > 1), and the convergence radius R of χ is bounded byR ≥ sup r > 0 : Br (0) ⊆ Ω(f [±1] )6DMITRII KOUZNETSOV AND HENRYK TRAPPMANNProof. By the equation χ f [n] (z) = K n χ(z) we get a rule to continue χ to Ω(f ±1 ):χ f [±n] (z)χ(z) =.
K ±n

We saw that we can express a super-function F of f through the inverse Schröderfunction: F (z) = η(sekz ). As it is mentioned in the introduction, there is wide
range of other a 7→ d super-functions: If F is a holomorphic a 7→ d super-functionof f then for every holomorphic 1-periodic function θ with θ(c) = 0 the function
E(z) = F (z +θ(z)) is another holomorphic a 7→ d super-function of f . Therefore, acondition, that singles out the solutions η(seκz ), would be quite welcome. Such a
condition is given in the next two propositions.
Proposition 6. Let κ be a complex number with <(κ) 6= 0. Then for each s ∈ C
the function F (z) = seκz is the only 0 7→ s solution of F (z +1) = eκ F (z), that is
holomorphic and bounded on some strip Sx,ε = {z/κ+x−ξ : 0 ≤ ξ < 1+ε, <(z) = 0}
(which is the strip between the straight lines orthogonal to κ going through the points
x and x−1−ε).
Proof. First F (z) = seκz is indeed a 0 7→ s solution and it is bounded on w ∈ Sx,εfor any fixed x and ε > 0 because |seκw | = |s| e<(z+κ(x−ξ)) = |s| e<(κ)(x−ξ) ≤|s| e|<(κ)|(1+ε+|x|) .
F1 (z) := eκz is also bounded from below by |F1 (z)| ≥ e−|<(κ)|(|x|+1+ε) . Let F̃be another solution, consider θ(z) = F̃ (z)/F1 (z) which must be bounded on Sx,ε .
Then θ(z+1) = θ(z), θ is a periodic function on Sx,ε . The translations of the strip
Sx,ε by integers k overlappingly cover C, ∪k∈Z (Sx,ε + k) = C. (Note that this is not
the case for <(κ) = 0.) Then, function θ can be holomorphically continued to C by
θ(z +k) = θ(z). Now θ is entire and bounded and hence constant: F̃ (z) = sF1 (z),where s must be F̃ (0).

The following proposition characterizes the solutions η(seκz ) as those which satisfy a certain asymptotic property in the direction κ of the complex plane.
Proposition 7. Let f be analytic at 0, f (0) = 0, K := f 0 (0), and |K| 6= 0, 1, letκ be some number with eκ = K; and let Hκ,x := {z/κ : <(z) ≤ x} be the halfplanebounded by the straight line going through and being orthogonal to x/κ. Then theset of on any Hκ,x holomorphic and bounded super-functions F of f which satisfy
limx→∞ sup |F (Hκ,−x )| = 0 consists exactly of the functions(2.22)

F (z) = η(seκz ),s∈C,where η is the inverse Schröder powerseries of f .
Proof. First, each F (z) = η(seκz ) is clearly a holomorphic super-functions of f .
It is bounded, because seκz maps Hκ,x to the closed disk |z| ≤ |s| ex , where η isholomorphic (continuous) and hence bounded.
Now let F be an arbitrary on Hκ,x holomorphic and bounded super-function of fwhich satisfies limx→∞ sup |F (Hκ,−x )| = 0; consider the function g(z) = η −1 (F (z)).
Choosing x small enough, by limx→∞ sup |F (Hκ,−x )| = 0 we can ascertain that
F (z) has values only within the convergence disk of η −1 for z ∈ Hκ,x . So g isholomorphic, and bounded on Hκ,x , and satisfies
g(z+1) = η −1 (f (F (z))) = Kη −1 (F (z)) = Kg(z)SUPER-EXPONENTIALS TO BASE

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7

for all z such that z, z +1 ∈ Hκ,x . As |K| 6= 1, we know that <(κ) 6= 0; and fromproposition 6 it follows, that g(z) = seκz and hence F (z) = η(seκz ).

Now we switch from the fixed point 0 to an arbitrary fixed point λ. If λ is a
fixed point of h, then 0 is a fixed point of z 7→ h(z +λ) − λ. It is clear, that, if χ
is a Schröder function (meaning that is satisfies the Schröder equation) of h, thenz 7→ χ(z+t) is a Schröder function of z 7→ h(z+t) − t for any t;χ h(z+t) − t + t = χ h(z+t) = Kχ(z+t)
and vice versa: if η is an inverse Schröder function of h, then z 7→ η(z) − t is aninverse Schröder function of z 7→ h(z+t) − t:
η(Kz) − t = h η(z) − t = h((η(z) − t) + t) − t .
Definition 3 (regular and principal super-function). Let h be analytic at its fixed
point λ and eκ = h0 (λ), <(κ) 6= 0. We call the functions z 7→ η(seκz )+λ, s ∈ C, theκ-regular super-functions of h at λ, where η is the inverse Schröder powerseries ofz 7→ h(z+λ) − λ. We call the function z 7→ η(eκz ) + λ the κ-principal super-functionof h at λ. Omitting κ above means κ = log(h0 (λ)) for the standard branch of thelogarithm.
Corollary 8. Every κ-regular super-function F is periodic with period T = 2πi/κand F (z) is a translation of the κ-principal super-function F(z) along the z-plane,i.e. F (z) = F(z +t) for some t ∈ C. We can always choose t = log(s)/κ, where s istaken from the previous definition.
Corollary 9. Let h be analytic at λ, h(λ) = λ, K := h0 (λ), such that |K| 6= 0, 1,
let κ be some number with eκ = K; and let Hκ,x as in proposition 7. Then theset on some Hκ,x holomorphic and bounded super-functions F of h which satisfy
limx→∞ sup |F (Hκ,−x ) − λ| = 0 is the set of the κ-regular super-functions of h at
λ.
Let us now consider strictly increasing real-analytic super-functions. We knowthat the inverse Schröder function of a real-analytic function h is strictly increasing
in a vicinity of the fixed point because η 0 (0) = 1, and we want to define the superfunction as F (x) = η(seκx ) + λ.
There is a natural distinction between positive and negative s. (We exclude thetrivial case s = 0, which gives the constant function F (x) = λ). Different s with thesame sign just translate the graph of F along the x-axis; ± |s| eκx = ±eκ(x+ln|s|/κ) .
The change of sign of s is more dramatic; it corresponds to a translation along theimaginary axis and inverses the monotony. We use this for a classification of theregular super-functions of real base-functions:Definition 4 (super-function above and below). Let h be real-analytic at the fixed
point λ = h(λ) such that 0 < h0 (λ) 6= 1. Let κ = ln(h0 (λ)). Let η be the inverse
Schröder powerseries of x 7→ h(x+λ) − λ. Then, we call the function
F (z) = η(seκz ) + λsuper-function of h below λ, for each s < 0; super-function equal to λ (i.e. theconstant function F (z) = λ), for s = 0; and super-function above λ, for each s > 0.

8

DMITRII KOUZNETSOV AND HENRYK TRAPPMANNLet I be the maximal interval around λ such that η is analytically continuable
to I and strictly increasing there. For any d ∈ η(I) + λ ands = η −1 (d−λ)(2.23)
we call the function(2.24)

F (z) = η(seκz ) + λthe regular 0 7→ d super-function F of h at λ.
The reader may verify that these functions are indeed all possible real-analytic
regular super-functions.
Let F (x) = η(seκx ) + λ, i.e. as in the previous definition. Let the word “finally”
mean: “there exist an x0 such that for all x > x0 ” in the case κ < 0, and “there exist
an x0 such that for all x < x0 ” in the case κ > 0. Then we can apply our previous
propositions and state:(1) For all ε > 0 finally |F (x)−λ| < ε.
(2) Finally F (x) > λ if s > 0.
(3) Finally F (x) < λ if s < 0.
(4) Finally F is strictly increasing if κs > 0.
(5) Finally F is strictly decreasing if κs < 0.
(6) F is holomorphic on a right halfplane if κ < 0.
(7) F is entire if κ > 0.
(8) F is periodic with the imaginary period T = 2πi/κ.
Actually the conditions 1, 6 and 8 already determine F to be a regular superfunction, as we show in the next proposition for the case κ < 0. This is a supplemental uniqueness criterion to the one given in proposition 7 and corollary 9.
Proposition 10. Let h be a function, real-analytic in a vicinity of its fixed point
λ, with 0 < h0 (λ) < 1; let κ = ln(h0 (λ)). Each real-analytic super-function F
holomorphic on a right half-plane Hx = {z ∈ C : <(z) > x} and 2πi/κ periodic thatsatisfies limx→∞ F (x) = λ is a regular super-function.
Proof. Let G(z) = η(eκz )+λ be the principal super-function of h. It is finally either
strictly increasing or strictly decreasing, hence finally injective. As limx→∞ F (x) =λ there is an x0 such that
(2.25)

W (x) = G−1 (F (x))

is well-defined for x > x0 . It satisfies(2.26)W (x + 1) = G−1 (h(F (x)) = W (x) + 1

and we can express F with G and W for x > x0 :(2.27)

F (x) = G(W (x)) = η(eκW (x) ) + λ

Now we have the precondition that F is 2πi/κ periodic. We know also that η(x)is injective in a vicinity ε of 0. Hence we possibly increase x0 somewhat so thateκx < ε for x > x0 . We apply then the injectivity of η on the periodic equality:η (exp (κW (x + 2πi/κ))) + λ = η (exp (κW (x))) + λexp(κW (x + 2πi/κ)) = exp(κW (x))κW (x + 2πi/κ) = κW (x) + 2πiN

SUPER-EXPONENTIALS TO BASE

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9

for some integer N . So W has the property


2πi
2πi
W x+
(2.28)= W (x) +NκκBy (2.26) θ(x) = W (x) − x is 1-periodic. It has a Fourier-development, whichmeans that there is a Laurent series ρ at 0 such that
W (z) − z = θ(z) = ρ e2πiz
W (z) = ρ e2πiz + z.
Now we plug this into equation (2.28) and derive



2πi
2πi
= ρ e2πiz + z +Nρ e2πi(z+2πi/κ) + z +
κ
κ


 2πi
2
(N − 1)ρ e2πiz e−4π /κ = ρ e2πiz +
κ


2
2πi
ρ ze−4π /κ = ρ(z) +(N − 1)κAs the Laurent series development is unique we obtain the equations2

ρn e−4nπ /κ = ρn
for n 6= 0 which implies ρn = 0 andρ0 = ρ0 +2πi(N − 1)κwhich implies N = 1. Hence ρ(x) = ρ0 is constant, then W (x) = x + ρ0 , thenF (x) = G(x + ρ0 ), which actually means that F is a regular super-function of h,see definition 3.
In the sequel we apply the propositions and statements before to h = exp√2 withthe two fixed points λ = 2, 4.
3. Super-exponential below fixed point 2
For base b > exp(1/e) a super-exponential can be evaluated through the Cauchy
integral, that gives the integral
equation for values of the function along the imag√
inary axis [11]. For b = 2, this algorithm cannot be applied as is, because both
fixed points λ = 2 and λ = 4 are real; but regular iteration can be applied for precise
evaluation.
√
Throughout this section, the base is b = 2, h = expb ; we consider the fixed point
λ = 2, so,
√ 
(3.1)
K2 = h0 (λ) = exp√2 0 (2) = 2 ln 2 = ln(2) ≈ 0.69314718055995
(3.2)
κ2 = ln(K2 ) = ln ln(2) ≈ −0.36651292058166 .
With formula (2.15) we compute the coefficients of the inverse Schröder function(3.3)v(z) = z +

∞
X
n=2

vn z n10

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

Table 1. Evaluation of coefficients v and V in (3.3), (3.18)
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

ofh[2](z):=(3.4)

h[2](z)

=

(3.5)

h[2]1

=

exp√2 (z+2) − 2 = 2∞
Xln(2)n2n−1 n!
n=1





ln(2)
exp
z −1
2

zn,

ln(2),
ln(2)2(3.6)
h[2]2 =
,
4
3
ln(2)
(3.7)
,
h[2]3 =
24and so on. As example we compute v2 and v3 via (2.18) and (2.19):
(3.8)

v2 =
v3 =

ln(2)/4ln(2)−1ln(2) (2+ln(2))/24(ln(2)−1)(ln(2)2 −1)
2≈ −0.56472283831773236365≈

0.33817758685118329988

More values for the coefficients v are shown in Table 1.
The regular 0 7→ 1 super-exponential is then by definition 4
(3.9)
F2,1 (z) = v(s2,1 eκ2 z ) + λ = v s2,1 ln(2)z + 2(3.10)

s2,1 = V (1−λ) = V (−1) ≈ −0.63209866105083

where V is the inverse powerseries of v (i.e. the Schröder powerseries of h[2]).
Because κ2 is negative the function F2,1 (z) is holomorphic on some right halfplane(see the statement list at the end of section 2).

SUPER-EXPONENTIALS TO BASE
y

√

2

11

y
q=0

8

8

6

6
p=2

4

4

2

2
q=0

0

A2 (z) ≈ 16

A2 (z) < 1

0

-2

-2

-4

-4

p=2

-6

-6

-8

-8

q=0

-8

-4

0

4

8

x

-8

-4

0

4

8

x

Figure 1. Function f = F̃2,1,20 (z) by (3.11) in the z = x+iy plane:levels p = <(f ) =const and levels q = =(f ) =const are shown withthick lines for integer values of p and q; left. Agreement f = A2 (z)by (3.12) in the z = x+iy plane; levels f = 1, 4, 8, 12, 16 are shownwith thick lines; right.
The truncation of the series (3.3), keeping the last term at index N gives afunction, let us call it F̃2,1,N , which approximates F2,1 (z) at large positive valuesof <(z):(3.11)

F̃2,1,N (z) = 2 +NX

vn sn2,1 ln(2)zn

n=1

For N = 20, this function is shown in the left hand side picture of figure 1 with lines
of constant real part and those of constant imaginary part. In the figure, the levelp = 1 visually goes through the origin of coordinates; in reality F̃2,1,20 (0) slightly
deviates from unity.
The precision of the approximation can be characterized with the agreement
F̃2,1,N (z)−logb F̃2,1,N (z+1)
A2 (z) = − lg
(3.12)
 ,F̃2,1,N (z)+logb F̃2,1,N (z+1)
Roughly, this agreement indicates, how many significant figures we may expect to
get with this approximation. In the left hand side of the picture, the agreement issmaller than unity; even the first decimal digit of the approximation is doubtful.
In the right hand side, the agreement is of order of 16. We expect, at <(z) > 3
the precision of approximation F̃2,1,20 (z) is limited mainly by the rounding errors.
Even in vicinity of the origin of coordinates, the approximation F̃2,1,20 gives several
correct decimal digits.
To evaluate F2,1 (z) for various values z, the integer iterations can be used:(3.13)[k]

F (z−k) = logb (F (z))

For the complexhdoublei implementation of F2,1 , we suggest the approximation
F̃2,1,20 (z)  , <(z) > 3(3.14)
F̃2,1 (z) =logb F̃2,1 (z+1) , <(z) ≤ 3
This approximation is estimated to return at least 14 significant decimal digits; it isused to plot function F2,1 in figure 2 in the same notations, as in figure 1. In the

12

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

p=3

y

q =0

p = 2.2

x

q =0

8

F2,1 (z)

6

p=2

p=4 4

q = 0.2
p = 1.8

2

q =0

8

4

p = −2q = 0.2

2

cut 0

q =0

-2

0

q =0

q = −0.2
p=2

-6

-4

p = −2-6

p = 2.2
q =0

-8

q =0
-8

-4

0

4

-8

q =0

F2,3 (z)6p=44q = −0.2
p = 3.8

p=2

q =0

p=2

-8

-4

0

4

q =0

−1
F2,3
(z)

cut

cut

4

0

p = 1.4
-4

p=4

p=2

4

q = 0.2
p = 3.8

p = 2.2

q =0
p = 3.8

0

4

8

p=2

-6

-8

q =0 y

-4

0

4

8

p=0

4

0

q = −8.6

cut

cut
q = 8.6

p=0

-4

p=1

∞

x

-8

q =0

F4,5 (z)

6

p=2

p=4 4

q = 0.2
p = 1.8
q =0

2

∞

q =0

q =9
p = 0.4-8

p = 3.8

8

cut
cut
cut
cut
cut

−1
F4,3
(z)

6

-6

p = 1.8

-8

x

p=1

-2

q = −0.2

-4

p=4

-4

8

2

q =0 0q = 8.4
q = 8.5
q = 8.55

yF4,3 (z)

6

-8

-8

∞

8

-2

p=2

-6

q = −0.2

-8

-4

y

0

4

8

x

p=4

8

−1
F4,5
(z)

6
4
2
0
-2

p = 4 -4q =0

q = −8.56x8q =0 y08 q = −8.55-6

p = 1.8

-8

q =0

8

q = 0.2

q = 0.2

-6

-2

4

-2

p=4

2

0

p = 1.4

-2

cut

-4

2

p = 2.2

q =0 0-4

p = −2.8

q =0

q =0
p = −3p = −3p = −3p = −3yy q = 8.55

82

p = −2.2

-8

x

8

y

cut

cut

p = −1.8
q = −0.2-2

p = 4-4

−1
F2,1
(z)

p = −2.2

6

p=3
q =1

cut

q =0

q = −1

p=3

-4
-6
-8

-8

-4

0

4

8

x

q =0

p=4
-8

-4

0

4

8

x

Figure 2. Functions f = F2,1 (z) , f = F2,3 (z) , f = F4,3 (z) ,f = F4,4 (z) , left, and their inverse functions, right, in the z = x+iy
plane, in the same notations as in fig.1.

SUPER-EXPONENTIALS TO BASE

√

2

13

y
y = F4,5 (x)864y = F2,3 (x) and y = F4,3 (x)
2
−8

−6

−4

−2

y = F2,1 (x)y = 1024 w(x)
2
4
6

8

x

−2

Figure 3. F2,1 by (3.9), F2,3 by (5.1) (which coincides with F4,3
by (5.2) with 24 significant figures), and F4,5 by (4.9). The asymptotes are shown with thin lines. The difference w by (5.11) scaledwith factor 1024 is shown with the oscillating curve in vicinity of
the abscissa axis.

right hand side of the plot, the function coincides with the primary approximation
F̃2,1,20 . The approximation also reproduces the periodicity of F2,1 :(3.15) F2,1 (z+T2 n) = F2,1 (z) , T2 =2πi
≈ −17.1431481793548471i , n ∈ Z .
κ2

The function F2,1 (z) has a branchpoint with logarithmic singularity at z = −2;
we place the cut along the line z < −2. Due to the periodicity, there is a set of
singularities and cut lines translated with period T2 by (3.15). The other picturesin the figure 2 represent functions considered in the following sections.
In figure 3 we plot the function F2,1 versus real argument; see the thick curveat the bottom. The other curves correspond to the super-exponentials which we
define and analyze in the following sections. We indicate the fixed point λ as firstsubscript of function F , and its value at zero as the second subscript. We choose
this value as the natural number, closest to the fixed point. In such a way, we deal
with the following super-exponentials:
F2,1 , developed at λ = 2, with range of values (−∞, 2) along the real axis,
F2,3 , developed at λ = 2, with range of values ( 2 , 4) along the real axis,
F4,3 , developed at λ = 4, with range of values ( 2 , 4) along the real axis,
F4,5 , developed at λ = 4, with range of values ( 4 , ∞) along the real axis,
and their inverse functions. All these functions are plotted in figure 2; their behavior
along the real axis is shown in figure 3.

14

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

3.1. The inverse super-exponential below 2. The inverse function of F2,1 (z) =v(s2,1 eκ2 z ) + λ is given by−1F2,1(z) = log (V (z − λ)/s2,1 ) /κ2(3.16)√ z
where V is the Schröder powerseries of h[2](z) := h(z + 2) − 2 = 2( 2 − 1). The−1function F2,1
is shown in the top of the right hand side of the figure 2. We calculate
the first few coefficients according to formula (3.7) and (2.15):(3.17)

V1 =
V2 =

1
ln(2)
−v2 = 41 1−ln(2)≈ 0.56472283831773236365

V3 =ln(2)2
1+2 ln(2)24 (1−ln(2))2 (1+ln(2))≈ 0.29964618138408807683

More values are printed in the right column of the table 1.
The series can be truncated, keeping the term of the N th power, giving theapproximation
!
N
X
1
1
−1
(3.18)
F̃2,1,N(z) =Vn · (z−2)n .
log
κ2s2,1 n=1
It is useful at small values |z−2|. The range of approximation can be extended
with the function(−1F̃2,1,N(z), |z−2| ≤ ρ−1

(3.19)
F̃2,1 (z) =−1
F̃2,1 expb (z) − 1 , |z−2| > ρ−1
In particular, for N = 20 and ρ = 0.4, the approximation F̃2,1
returns of order of 14
correct decimal digits and can be used for the complexhdoublei implementation innumerical analysis.
−1
The plots of functions F2,1 and F2,1
look similar. This similarity indicates approximate symmetry x ↔ −y of the graph y = F2,1 (x); this function is shown again
in left hand picture of the figure 7; indeed, it looks pretty symmetric. We analyze
this visual effect in section 8.

4. Super-exponential above fixed point 4
In this section is√b = 2,

h(z) = expb (z),

(4.1)

λ=4
√ 
K4 = h0 (λ) = exp√2 0 (4) = 4 ln 2 = 2 ln(2) ≈ 1.3862943611199(4.2)

κ4 = ln(K4 ) ≈ 0.32663425997828098238.

With formula (2.15) we compute the powerseries u of the inverse Schröder function of h[4](z) := h(z+4) − 4




ln(2)
(4.3)
h[4](z) = 4 exp
z − 1 = 2h[2](z)2∞Xln(2)n n(4.4)
h[4](z) =z
2n−2 n!
n=1(4.5)

h[4]1 = 2 ln(2), h[4]2 =ln(2)2ln(2)3, h[4]3 =
,...
2
12

√

SUPER-EXPONENTIALS TO BASE

2

15

Table 2. Coefficients u and U in expansions (4.9) and (4.22)
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

(4.6)

un
1.0000000000000000
0.4485874311952612
0.1903722467978068
0.0778295765369683
0.0309358603057080
0.0120221257690659
0.0045849888965617
0.0017207423310577
0.0006368109038799
0.0002327696003030
0.0000841455118381
0.0000301156464937
0.0000106807458130
0.0000037565713616
0.0000013111367785
0.0000004543791625
0.0000001564298463
0.0000000535232764
0.0000000182077863
0.0000000061604765

Un
1.0000000000000000−0.4485874311952612
0.2120891200549197−0.1021843675069717
0.0496986830373718−0.0243075903261196
0.0119330883965109−0.0058736976420089
0.0028968672871058−0.0014309081060793
0.0007076637148566−0.0003503296158730
0.0001735756004664−0.0000860610119291
0.0000426959089013−0.0000211930290682
0.0000105244225996−0.0000052285174354
0.0000025984499916−0.0000012917821121

ln(2)/4
2 ln(2)−1

u2 =≈ 0.44858743119526122890



(4.7)

u3 =

1+ln(2) ln(2)2 /12
1−2 ln(2)−4 ln(2)2 +8 ln(2)3≈ 0.19037224679780675668

More values for the coefficients u are printed in table 2.
While in the case λ = 2 the preferable initial condition is F2,1 (0) = 1, which
implies√2√ ..
F2,1 (n) = exp√2 [n] (1) = 2,| {z }
√
n× 2
.

(4.8)

we see no so evident preference for λ = 4. So we choose F (0) = 5 for the superexponential above 4. From (2.22), for η = u, we get(4.9)

F4,5 (z)(4.10)s4,5= u(s4,5 eκ4 z ) + 4= U (5−4) = U (1) ≈ 0.69470714714324 .

In analogy with v and V , let U denote the inverse function of u; and let F4,5
denote the corresponding super-exponential. The choice F4,5 (0) = 5 allows theinterpretationof F4,5 (n) as result of n applications to number 5 of the exponential√
to base 2:(4.11)

F4,5 (n) = exp√2 [n] (5) =

√ ...
2
|
{z

√ 5
2

,
}

n repetitions of

√

2

16

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

q=0 y

y

8

8

6

6

4

4

p=4

2

2

q=0 0

0

-2

-2

-4

-4

-6

-6

-8

-8

A4 (z) ≈ 16

A4 (z) < 1

p=4

q=0

-8

-4

0

4

8

x

-8

-4

0

4

8

x

Figure 4. Left: Function f = F4,5 (z) in the x+iy plane; levels
p = <(f ) =const and levels q = =(f ) =const are shown with thicklines for integer values of p and q. Right: Function f = A4 (z) by(4.13) in the x+iy plane; levels f = 1, 4, 8, 12, 16 are shown withthick lines.
at least for natural numbers n.
Note, that the super-exponentials below 2, for example F2,1 , remains smaller
than 2 along the real axis, actually in the interval (−2, ∞), as we did not define
it for (−∞, −2]; but the super-exponentials above 4, for example F4,5 , remains
larger than 4 along the whole real axis. The super-exponential below 4 will beconsidered later together with the super-exponential above 2; along the real axis,
these functions are also plotted in figure 3.
For the evaluation of (4.9) and (4.10), in analogy with super-exponential developed at λ = 2, we truncate the powerseries. The truncation of the powerseries u,
keeping N +1 terms, leads to the following approximation, let us call it F̃4,5,N , ofthe super-function F4,5 :(4.12)

F̃4,5,N (z) = 4 +NX

un sn4,5 exp(nκ4 z) ;n=1

The function F̃4,5,20 is plotted in the complex z-plane in the left hand side pictureof the figure 4. In the right part of the left picture, the density of the levels is sohigh, that the plotter cannot draw them; this part is left empty.
In order to characterize the precision of the approximation of solution F4,5 ofequation (1.1), in the right hand side picture of figure 4, the agreement
F̃4,5,20 (z)−expb F̃4,5,20 (z−1)
A4 (z) = −lg(4.13)
F̃4,5,20 (z)+expb F̃4,5,20 (z−1)
is plotted. The plot indicates, that, at <(z) < −1, the error of evaluation is determined by the rounding errors of the complexhdoublei arithmetic, not by the error of
the approximation F̃4,5,20 . For the approximation of the solution F4,5 in the wholecomplex plane, we suggest the function
F̃4,5,N (z)  , <(z) < xN
(4.14)
F̃4,5 (z) =expb F̃4,5 (z−1) , <(z) ≥ xN

SUPER-EXPONENTIALS TO BASE

√

2

17

For N = 20 and x20 = −1 it returns of order of 14 significant figures of function
F4,5 . This approximation is used to plot F4,5 in figures 2 and 3. The evaluation
can be performed also at smaller values of N ; then, for the good precision we needto choose |xN | > |x20 |.
In the left hand side of the complex plane, F4,5 approaches the fixed point λ = 4.
Along the real axis, the function grows up faster than any exponential, and it shows
the fractal behavior in vicinity of the positive part of the real axis. At moderated
distances from the real axis, the function F4,5 changes slowly, approaching the fixed
point λ = 2.
The function F4,5 is periodic; the period T4 is expressed by(4.15)

T4 =2πi≈ 19.236149042042854712 i ,κ4so, at the distance of order of 19 from the real axis, the fractal structure is reproduced, although it is out of range of its plot in figure 2. The function shows rapid
growth in the positive part of the real axis and complicated behavior in its vicinity,
but it is analytic in every point of the complex plane:
Proposition 11. F4,5 is entire.
Proof. h = exp√2 is entire and K = 2 ln(2) > 1. So proposition 4 can be applied.
However, the growth of the entire F4,5 in vicinity of the real axis is so rapid,that even in the range of the plot in figure 2, the values cannot be stored in acomplexhdoublei variable; there values are marked with symbol ∞. Such a rapid
growth is typical for the growing super-exponentials.
−1
4.1. Inverse super-exponential above 4. The inverse function F4,5of function
F4,5 can be defined with relation
−1(4.16)
F4,5 F4,5(z) = z ∀z ∈ C\{x ∈ R : x ≤ 4} ,

−1
(4.17)= F4,5(z)< =(T4 )/2 ≈ 9.6180745210214273558

It is shown at the bottom of figure 2. The range of validity of the reciprocal relation
−1(4.18)
F4,5
F4,5 (z) = z
is narrower, than C\{x ∈ R : x ≤ 4}; it is limited by a single period of function−1
F4,5 along the imaginary axis. The function F4,5(z) has only two singularities at
z = 2 and z = 4, we choose the cut line along the real axis, z < 4. In the positive
direction of the real axis, the function slowly grows, slower than a logarithm. We
consider the properties of this function and its evaluation in this section.
The inverse function satisfies the Abel equation [1, 3]−1−1(4.19)
F4,5
expb (z) = F4,5(z) + 1This function can be expressed with the Schröder powerseries U of h[4]:(4.20)−1F4,5(z) = log(U (z−4)/s4,5 ) /κ4

18

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

We compute the first some coefficients of U with formula (2.15)(4.21)

U1 =
U2 =

1
ln(2)/4−u2 = 1−2ln(2)≈ −0.4485874311952612289

U3 =(4 ln(2)+1)/24
1−2 ln(2)−4 ln(2)+8 ln(2)2≈ 0.21208912005491969757

More values are printed in the second column of table 2. Numerically, the inverse
function can be implemented with
!
N
1 X
1
−1
n
(4.22)
log
Un ·(z−4);F̃4,5,N (z) =κ4s4,5 n=1
this approximation is useful for small values of |z−4|. The range of the approximation can be extended with function
(

−1
F̃4,5
logb (z) +1 , |z−4| > ρ−1(4.23)
F̃4,5 (z) =−1
F̃4,5,N(z), |z−4| ≤ ρ
At N = 20 and ρ = 0.4, such approximation returns at least 14 significant figuresand can be recommended for a complexhdoublei implementation.
5. Super-exponentials above 2 and below 4
In √this section, we consider the two remaining classes of super-exponentials tobase 2. These are exemplary given by(5.1) F2,3 (z) = v(s2,3 eκ2 z )+2 ,(5.2) F4,3 (z) = u(s4,3 e

κ4 z

)+4 ,

s2,3 = V (3−2) =V (1) ≈

2.1844747586390

s4,3 = U (3−4) = U (−1) ≈ −1.8604041940071

s is chosen such that F2,3 (0) = F4,3 (0) = 3.
Every regular super-exponential F (z) at fixed point λ is a translation in thez-plane of the principal (see definition 3 and corollary 8) super-exponential. If s is
positive it is a real translation otherwise it is a complex translation. The superexponential with s = −1 can be obtained by a translation along the imaginary axis
by the half of the corresponding period T = 2πi/κ:−eκz = eκz+πi = eκ(z+πi/κ) = eκ(z+T /2) .
For systematization of our super-exponentials, let us begin with the case, when,in (2.22), the Schröder parameter s is unity. We call the corresponding superfunction “principal super-exponential” or, if we need to indicate value of κ, “theκ-principal super-exponential”. Let Fλ denotes the principal super-exponential atλ, id est, Fλ (z) = η(exp(κλ z)) + λ; we use this expression for functions η = u
and η = v. Then, all our super-exponentials can be expressed as principal superexponentials with displaced arguments:
ln |s2,1 |(5.3)≈ 1.25155147882219
F2,1 = F2 (z+t2,1 +T2 /2)
t2,1 =
κ2ln(s2,3 )(5.4)
F2,3 = F2 (z+t2,3 )
t2,3 =≈ −2.13191778709503
κ2
ln |s4,3 |(5.5)
F4,3 = F4 (z+t4,3 +T4 /2)
t4,3 =≈ 1.90057764535872
κ4
ln(s4,5 )(5.6)
F4,5 = F4 (z+t4,3 )
t4,5 =≈ −1.11520724513161
κ4

SUPER-EXPONENTIALS TO BASE

y

y

8

8

6

6

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8
-8

-4

0

4

8

x

-8

√

2

-4

19

0

4

8

x

Figure 5. Functions w and w̃ by (5.11) in the complex plane
In consequence the functions F2,3 and F4,3 correspond to a complex translation of
the functions F2,1 and F4,5 , respectively:(5.7)

F2,3 (z) = F2,1 (z+t2,3 −t2,1 −T2 /2)(5.8)

F4,3 (z) = F4,5 (z+t4,3 −t4,5 −T4 /2)−1−1
We place the cut lines of F2,3
and F4,3
along the half-lines line <(z) < 2 and<(z) > 4, in order to keep them analytic in vicinity of z = 3. F2,3 and F4,3 areshown in the left hand side of figure 2 and their inverse are plotted in the righthand side.
The plots for F2,3 and F4,3 look similar. At least for =(z) < T2 /2, the two(0 7→ 3) super-exponentials are related with


−1
(5.9)
F2,3 (z) = F4,3 z + θ2,4 (z) ,θ2,4 (z) = F4,3
F2,3 (z) − z ,where θ2,4 is a 1-periodic function. At least in vicinity of the real axis, it can beapproximated with the sinusoidal
(5.10)
θ2,4 (z) ≈ θ̃2,4 (z) = ξ · sin(φ+2πz) − sin(φ)
with parameters ξ = 0.7252175546·10−24 and φ = 0.74633983476 . The approximation θ̃2,4 can be used for the complexhdoublei implementation of w(z) = F4,3 (z)−F2,3 (z) for |=(z)| < 4 :(5.11)

w(z) = F4,3 (z)−F2,3 (z) ≈ w̃(z) := θ̃24 (z) · F4,3 0 (z)Function w is plotted in the left hand picture of figure 5, and function w̃ in theright hand picture. The plots of w and w̃ look identical at moderate values of
the imaginary part of the argument. In addition, each of them is zero, while theargument is integer. However, the entire function w̃ reproduces neither the cutlines,
nor singularities of function w; and the deviation of w̃ from w becomes visible atthe distance of order of |T2 | /2 ≈ 8.5 form the real axis.
Any small periodic or quasi-periodic perturbation of a holomorphic function, realalong the real axis, grows up in the direction of imaginary axis. The quasi-period
unity determines the increment of this growth; is of order of 2π. In the case offunction w, this perturbation becomes of order of unity at the distance of orderof |T2 | /2. This allows to estimate the smallness of the deviation w in vicinity of20

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

the real axis. At the distance of order |T2 | /2 from the real axis, the quasi-periodic
perturbation of the function grows up until unity (and even becomes infinite at
the singularities of F2,3 ). This gives an estimate of the order of magnitude of thisdeviation; for moderate values of =(z), id est, in vicinity of the real axis,


−2π 2
(5.12)|w(z)| < σ := exp(−π |T2 |) = exp|ln(ln(2))|≈ exp(−53.8) ≈ 4 · 10−24
The perturbation is not exactly sinusoidal. As we leave from the real axis, the highest harmonics grow faster; so, we may expect, that, at the real axis, the deviationis smaller than σ. The function 1024 w(x) versus real x is shown at the bottom of
figure 3; it oscillates in vicinity of the real axis and, indeed, for real x the values of|w(x)| are an order of magnitude smaller than σ.
In such a way, the order of magnitude of the deviation between the two (C, 0 7→ 3)
super-functions of expb is determined by their periods, which, in turn, is determined√by the derivatives of the exponentials at the fixed points λ = 4 and λ = 2 for b = 2.
The deviation D by (7.3) of the iterated exponentials, made of super-function
with periods T4 and T2 also is smaller than σ in the (2,4) interval; this deviation isconsidered in section 7. The deviation between two holomorphic functions cannot
remain small everywhere (while this deviation is not a constant); therefore it canbe seen comparing the maps of these functions in the complex plane. The extension
to the complex plane is a universal tool for the analysis of functions that differ only
little at the real axis.
6. Periodicity of inverse super-exponentials
When considering inverse holomorphic super-functions in the complex plane one
is tempted to make some mistakes of the same category as the branch-related
mistake log(ab) = log(a)+log(b) for a, b ∈ C \ {0}. One merely can state that foreach a and b we can choose a branch of the logarithm on the left side such that theprevious equation is true. Or one can restrict the validity of the equation to theright half-plane.
Without going too deep into the theory of global holomorphic functions, we wantto mention this source of confusion. Especially for derivations like:(6.1)
F (z+1) = h F (z)(6.2)
F −1 (z) + 1 = F −1 h(z)the inverse of a super-function may be an Abel-function only on some restricteddomain or up to choosing the correct branch.
Moreover for inverse real-analytic super-exponentials G the following derivationis correct only up to choosing the proper branch. Let T = 2πi/ ln(b) be the periodof z 7→ bz :(6.4)

G(bz ) = G(z) + 1
G(z) + 1 = G(bz ) = G bz+T = G(z+T ) + 1(6.5)

G(z) = G(z+T )(6.3)

SUPER-EXPONENTIALS TO BASE

√

2

21

Proposition 12. Let G be the inverse (taken along the real axis and analyticallycontinued) of a regular super-exponential to base b ∈ (1, e1/e ), then G is periodic(6.6)

G(z+T ) = G(z) ,

T =

2πi
ln(b)

up to choosing a suitable branch of G.
From the considered inverse super-exponentials — with cuts as indicated — the−1
only properly periodic function is F2,1
, i.e.
(6.7)

−1
−1
F2,1
(z+Tslog ) = F2,1(z) ,Tslog =

4πi
≈ 18.129440567308775239 iln(2)for each z 6= (2, ∞) + Tslog k, k ∈ Z.
The period Tslog is determined by the jump of the function F2,1 at the cut between
points −1 and −2; which is of the same order of magnitude as the period of thesuper-exponential F4,5 by (4.15) and the period T2 of the tetrational by (3.15). The
closeness of the periods√ T2 to the period Tslog seems to be a specific property ofthe tetrational to base 2. In particular, such a closeness allows functions F2,1 andx 7→ −F2,1 −1 (−x) to be nearly identical along the real axis; this topic is consideredin section 8.
7. Non-integer iteration of the exponential to base sqrt(2)−1
The pair Fλ,d and Fλ,dcan be used to define fractional iteration of the exponen√tiation to the base b = 2. For λ = 4, we define


[c]
−1
(7.1)
expb,4 (z) = F4,d c + F4,d(z) ∀z ∈ C\{x ∈ R : x ≤ 4} .

In particular, we may use d = 3 or d = 5 in (7.1).
Due to relation (5.8), the displacement of the argument of function Fλ,d is com−1
pensated by the displacement of value of of function Fλ,d. For this reason, we donot specify the value of d in the left hand side of (7.1); it does not depend on d.
[c]
For c = 0.5 the expb,4 by (7.1) is shown in the right hand side picture of figure 6
in the same notations as in figure 1.
However, the shape of the iterated exponential depends on the fixed point λ used−1
to construct the super-functions. The pair of functions F2,d and F2,dalso allows toconstruct the iterated exponential
(7.2)

[c]

−1
expb,2 (z) = F2,d (c + F2,d(z))

−1
in a similar way as is constructed with pair F4,d and F4,din equation (7.1). The
range of holomorphism is a little bit more complicated than in the case λ = 4 because−1of the complicated behavior of the F2,dshown in the right hand side of figure 2.
Similarly, due to relation (5.7), we can use any of the already described functions,[c]either d = 1 or d = 3, and get the same expb,2 . For c = 0.5, this function is plottedin the left hand side picture of figure 6.
[c][c]
The functions exp√2,2 and exp√2,4 versus real argument are plotted in the bottom, right picture of figure 6. In the range 2 < x < 4 these curves overlap. (Theywere expected to overlap, because, along the real axis, the functions F2,3 and F4,3
overlap with 24 significant figures, see figure 3).

22

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN
q= 0

cut
cut
cut
cut
cut
q = −9y8
642

q= 0 0

y

[1/2]
cut

cut 0

-6
-8
q= 0

0

4

8

x

y
8
6

D

2

[1/2]

q = 10expb,4q= 2
q= 0
q = −2-2-4 q = −0.2

4

p = 102

p = −1-2 p = −1.2

-4

p= 6

6
4

expb,2

-8

8

q= 9
cut
cut
cut
cut
cut

-4

q = −10-6

p = 10

-8

p= 6

-8

-4

0

cut
cut
cut
cut
cut

y

cuts

5

4

8

x

[0.5]

y = expb,4 (x)

4

cut 0

cut

3

-2

2
cuts

-4
-6
-8
-8

-4

0

4

8

x

cut
cut
cut
cut
cut

1
−6

−4

−2

2

4
x
y = 1024 D(x)[0.5]

y = expb,2 (x)

[c]

[c]

Figure 6. Comparison of functions expb,4 (top, left) and expb,2(top, right) for c = 0.5; difference D (left,bottom) in the complex
plane; and both functions versus real argument (right, bottom):[c][c]y = expb,4 (x) by 7.1, solid, and y = expb,2 (x) by 7.2, dashed; function D by (7.3), scaled with factor 1024 , oscillates in vicinity of
the abscissa axis, 2 < x < 4.
[0.5]

[0.5]

In spite of the similar values along the real axis, functions exp√2,4 and exp√2,2
[c]

[c]

are different. Function expb,4 (z) has a single cut-line at z < 2. Function expb,2 (z)has two sets of cut-lines<(z) > 4 , =(z) = =(Tslog )n , n ∈ N;
and



 2n+15<(z) > < F2,1 − ± oi
≈ 0.8386449382477 , =(z) = = Tslog, n ∈ N;22
also, this function has a fractal of singularities and additional cut-lines at <(z) > 4,
−1
due to the singular behavior of function F4,5(z) in that region.
[c]−1
Function expb,4 (z) is not periodic, as F4,5
is not periodic.
[c]−1
Function expb,2 (z) is periodic, as F2,1
is periodic; the period Tslog is determined by(6.7).
In order to characterize the deviation of the two iterated exponentials, we analyze
the difference

(7.3)

[0.5]

[0.5]

D(z) = exp√2,4 (z) − exp√2,2 (z) .

SUPER-EXPONENTIALS TO BASE

√

2

23

Function D, scaled with factor 1024 is plotted in the range 2 < x < 4 in the rightbottom part of figure 6; |D(x)| < σ.
The two iterated exponentials strongly deviate from each other in vicinity of[c]z = <(F2,1 (−2.5 + oi)) + Tslog /2, where the function expb,2 (z) has a singularity.
(Symbol oi indicates that the function should be evaluated just above the cutline.)
For |z−3| < 1, the function D(z) can be roughly approximated with the 6parametric fitD̃(z) = 2.48 · 10−25 (z−2)(4−z) 1 + 0.120(z−3) + 0.006(z−3)2 ×


(7.4)
sin 0.747 − 0.068(z−3) + 0.007(z−3)2 + T4 ln(4−z) + T2 ln(z−2)
The constants T2 and T4 are determined by (3.15) and (4.15). In figure 7, both
functions 1024 D and 1024 D̃ overlap well and the error of the approximation D̃ is notseen even at strong zooming-in. For detailed numerical analysis of the deviation
between the two (0 7→ 3) super-exponentials in vicinity of the real axis, even more
precise fit of function D can be elaborated on the base of approximation (5.11) ofdifference w between the two super-exponentials, treating the constant σ by (5.13)
as a small parameter.
As in the case of super-exponentials F2,3 and F4,3 and as in the case of the[c][c]iterated exponentials expb,2 and expb,4 precise evaluation is required to see thatthe two functions are not the same. Dan Asimov privately communicated that he,
Dean Hickerson and Richard Schroeppel recognized this deviation of the half-iteratearound 1991 by computing the values to 25 digits of accuracy. The only accessiblereference is his posting [8] on sci.math in form of an April fool’s joke.
However, the deviation becomes more visible outside the real axis, functions areclearly distinguishable without any zooming-in. In the following section we consider
an additional (and last) example when leaving the real axis significantly simplifies
the consideration.
8. About the apparent (x,-y) symmetry of the graph of thesuper-exponential below 2.
The (x, −y) symmetry of graph of y = f (x), apparently shown in figure 7 for
f = F2,1 , can be described by the formula(8.1)

f −1 (−x) = −f (x)or, applying function f to equation (8.1),
(8.2)
f − f (x) = −x .
Generally, for some function f , there is nothing wrong in relations (8.1), (8.2).
For example, each of functionsp
2x+3
f (x) = 1+x ,f (x) = 25−(3−x)2 − 3 , f (x) =x+2
satisfies (8.1), (8.2); and the graphic of the last of them even has the same verticaland horizontal asymptote x = −2 and y = 2 (although does not approach them
exponentially). In this section we show, that function f , satisfying (8.2), (8.1),
cannot be F2,1 .

24

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

2 y = F (x)
2,1

y
0.01

1
−2

−1

1

2

x

3

−2

−1

−1

−0.01

−2

−0.02

−3

−0.03

−1
y = F2,1 (x) + F2,1
(−x)

1

2

3

4

5

6

7

x

y = F2,1 (−F2,1 (x)) + x

Figure 7. Function y = F2,1 (x) of real argument left; functions−1
y = F2,1 (−F2,1 (x)) + x and y = F2,1 (x) + F2,1(−x)), right.
Below, we present two independent proofs. The first proof shows, that such asymmetry does not hold for a function with asymptotic behavior of function F2,1 .
The second proof shows more√generally, that such a symmetry cannot hold for any
super-exponential f to base 2 that is monotonous on the (−2, ∞) interval. Bothpropositions below and their proofs operate exclusively with real quantities.
Proposition 13. y = F2,1 (x) is not (x, −y) symmetric.
Proof. Here, we do not put indices on the function f , assuming that f = F2,1 . Weshow that for such function, equation (8.2) does not hold. The deduction is basedon the asymptotic properties of the function. Consider the behavior of function
f (x) in vicinity of the singularity at x = −2:f (−2+x) = logb (f (−1+x)) = logb f 0 (−1)x + O(x2 )(8.3)
In the right hand side of (8.3), the expansion of f begins with linear term, becausef (−1) = 0. Using notation(8.4)

Φ := f 0 (−1) =F2,1 0 (0)√  ≈ 1.69973893 ,ln 2

equation (8.3) can be rewritten as
(8.5)




f (−2+x) = logb Φx + O(x2 ) = logb (Φx)· 1 + O(x)For positive u, v, ln(uv) = ln(u)+ln(v); so, at least for positive x,f (−2+x) = logb (Φ) + logb (x) + logb 1 + O(x)
(8.6)

=

logb (x) + logb (Φ) + O(x) .

Now remember the asymptotic behavior of F2,1 (z) at large values of <(z); from
(3.9) and (3.3), we can represent f (x) as follows:(8.7)

f (x) = 2 − ω − O(ω 2 ) ,where ω = −s2,1 eκ2 x > 0. Then, consider the expansion at large positive x of theexpression



(8.8)
f − f (x) = f − 2 + ω + O(ω 2 ) .

SUPER-EXPONENTIALS TO BASE

√

2

25

Using the expansion (8.6) at x = ω + O(ω 2 ), we getf − f (x) = logb ω · (1 + O(ω)) + logb (Φ) + O(ω)
logb (ω) + O(ω 0 )
Substituting ω to exp (x+ln |s2,1 |)κ2 , for x → ∞ we get
=

(8.9)

(8.10)

f − f (x) =κ2 x
2 ln(ln(2))√ + O(x0 ) =x + O(x0 ) ;ln(2)ln( 2)

equation (3.2) is used to get the last equality. The coefficient(8.11)

2 ln(ln(2))≈ −1.057532745889795
ln(2)is close to minus unity, but not exactly minus unity. At large positive x, theexpression −f − f (x) deviates from x asymptotically linearly, with tangent oforder of 6%. This deviation is seen at the plot of y = f (−f (x)) + x in the righthand picture of figure 7.
The symmetry can be negated also without to refer the asymptotic analysis, justfrom the observation that f is a monotonous (increasing or decreasing) function.
The deduction based on the sequential application of hypothesis (8.2) is presented
below.
Proposition 14. There is no monotonous function f on (−2, ∞) that satisfies(8.12)

f (0) = 1
√ f (x)f (x+1) = 2(8.13)f (−f (x)) = −x(8.14)

Proof. Below we show that there exist 3 points x1 , x2 , x3 such that x1 < x2 < x3 ,but f (x1 ) < f (x2 ) > f (x3 ). No one monotonous function is allowed to have such aproperty.
√
Let us show that there are these 3 points x1 < x2 < x3 ; set b = 2. We startwith the more difficult part of finding x2 < x3 with f (x2 ) > f (x3 ). Applying (8.13),
we getf (0) = 1
f (1) = b
f (2) = bb =: c.
It follows further
f (4) = bbccf (−bb ) = −4cf (−bb + 1) = b−4
and we set(8.15)

x2 := −b−4cf (x2 ) = bb − 1.

26

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

On the other hand we conclude
f (−c) = −2
f (−c + 1) = b−2
f (−b−2 ) = c − 1
f (−b−2 + 1) = bc−1
f (−bc−1 ) = b−2 − 1and setx3 = −bc−1 + 1(8.16)−2f (x3 ) = bb

−1.

Now we compare x2 and x3
x2 = −b−4 = −0.25 < −0.24510159 ≈ −bc−1 + 1 = x3
and their valuescf (x2 ) = bb − 1 ≈ 0.84091087 > 0.84089642 ≈ bb

−2

−1

= f (x3 )x1 can then easily be found: say x1 = −1 < x2 then f (−1) = 0 < f (x2 ).



There is no need to depart from the real axis in order to see that the (x, −y)symmetry of the graphic in the left hand side picture of the figure 7 is only approximation, although in the range (−1, 0), this “symmetry” holds with 4 decimal
digits. Having the complexhdoublei implementation of function f , the left handside of the expression (8.2) can be precisely evaluated; the residual is plotted in theright hand picture of figure 7, confirming the approximate character of the (x, −y)“symmetry”.
We included both propositions above, because they are beautiful and could begood exercises for the first-grad students who do not yet deal with functions ofcomplex variable. However, the approximate character of the (x, −y) “symmetry”−1
follows directly from the periodicity of functions F2,1 and F2,1
. Both functions
are periodic, but their periods T2 by (3.15) and Tslog by (6.7) are not equivalent.
If two functions, holomorphic at some domain C, coincide at some interval, forexample, (−1, 0), then they coincide in the whole domain, and cannot have so
different periods, assuming that they are not constant.
We have considered three pairs of “similar” functions: (F2,3 , F4,3 ) by equations[0.5] [0.5](5.1),(5.2), shown in figure 3; exp√2,2 , exp√2,4 by equations (7.2),(7.1), shown in
−1
figure 6, and, en fin, F2,1 , x 7→ −F2,1(−x) by equations (3.9),(3.16), plotted in
the left hand side picture of figure 7. Here, the “similarity” means, that at the realreal plot, visually we cannot distinguish the function from its partner. In the lastcase (considered in this section), the difference between these functions is plottedwith dashed curve in the right hand side picture of figure 7; the float precision ofevaluation is sufficient to reveal the deviation. In the first two cases, of order of 30
decimal digits should be evaluated in order to reveal the deviation along the realaxis. In all examples considered, a high precision of evaluation is not required tosee the deviation in the complex plane.
The different periodicity (in our case, periods T2 , Tslog , T4 and ∞) allows to
reject the hypothesis of the coincidence of the two functions without to make any
numerical evaluation, without the asymptotic analysis of proposition 13, and without the algebraic deduction of proposition 14.

SUPER-EXPONENTIALS TO BASE

√

2

27

9. Summary and prospects
I. We have constructed four real super-functions Fλ,d of the base-function exp√2by (5.3)-(5.6). All these functions are periodic, their periods Tλ are pure imaginary
and are determined by the fixed points λ = 2 and λ = 4. F2,1 by equation (5.3)
is a tetrational. The functions are bijective on the real axis with the following(co)domains (see figure 3):
F2,1 : (−2, ∞) → (−∞, 2)

strictly increasing

F2,3 : (−∞, ∞) → (2, 4)

strictly decreasing

F4,3 : (−∞, ∞) → (2, 4)

strictly decreasing

F4,5 : (−∞, ∞) → (4, ∞)strictly increasing

II. The complex translations of the argument relates function F2,1 to function
F2,3 ; both function are holomorphic at least in the right hand side of the complex
plane. The complex translations of the argument relates function F4,3 to function
F4,5 ; both functions are entire. We describe the complex hdoublei implementationfor each of these functions and their inverses (see fig.2), and that for the deviationof F4,5 from F4,3 , see figure 5. In vicinity of the real axis, this deviation is of order−1of 10−24 . Functions F2,1 and its inverse function F2,1
have similar periods T2 by−1(3.15) and Tslog by (6.7); in the range −1 ≤ x ≤ 0, the deviation of −F2,1(−x) from−4
F2,1 (x) is of order of 10 , see figure 7.
[c][c]III. With functions Fλ,d , the two different iterated exponentials expb,2 and expb,4are constructed; they coincide at integer values of c, but for non-integer c, they havedifferent asymptotic properties and different ranges of holomorphism. For c = 0.5 weplot both functions in figure 6; in the range (2, 4) along the real axis, the deviation
between these functions is of order of 10−24 , and each of these two functions can√be arguably considered as a “true” exp√2 .
IV. The construction of the base-function h for a given super-function F is easierthan the construction of a super-function for a given base-function. In principle thebase-function is given by h(z) = F (1 + F −1 (z)). Such “inverse” method could be
used to build-up some “table of super-functions” as an analogy of a table of integrals. Up to our knowledge, no such table for super-functions has been published.
The compilation of such a table could be matter for future research.
V. The regular iteration of the base-function at a fixed point, can be used forconstruction of various real super-functions for real base-functions with real fixed
points. For example, there should√ exist at least two different holomorphic halfiterates of the factorial (id est,
! ); this function is used as logo of the Physics
Department of the Moscow State University since past century, without to implement an algorithm for the efficient evaluation. Only recently an algorithm forevaluation of such a function has been suggested [13].

Acknowledgment
Authors are grateful to the participants of the Tetration Forum [17] for theimportant questions and discussions.

28

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

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Mathematics of Computation, 57 (1991), 723–733.
17. http://math.eretrandre.org/tetrationforum/index.php
18. http://en.citizendium.org/wiki/Superfunction
19. http://en.citizendium.org/wiki/Tetration

Appendix A. Notations
symboldescription

A2
agreement for complex(double) implementation of F2,1
A4
agreement for complex(double) implementation of F4,5
a √reference point; for ex., F (a) = dbase of the super-exponential
b = 2 ≈ 1.414213562373
C
set of complex numbers
C⊆C
range of the holomorphism of the superfunction Fcnumber of iterations; c = 0.5 is used as exampleχ = η −1
general symbol for Schröderfunction
X (z) = sχ(x)any solution of (2.14) in Proposition 2
D⊆C
range of holomorphizm of function G
0.5
D(z) = exp0.5
difference of two generalized exponentials
b,4 − expb,2D̃(z)approximation of D(z) for |z − 3| < 1dvariable used in definition of the super-functionref.
(3.12)(4.13)
(1.2)
(1.4)
(1.1)
(7.2)
(2.13)
(2.3)
(7.3)
(7.4)
(1.2)

SUPER-EXPONENTIALS TO BASE

symbol√2

29

description

E(z) = F (z +Pθ(z))
super-function with modified argument∞e = exp(1) = n=0 1/n! ≈ 2.71
base of natural logarithms
exp(z) = ez
natural exponential
expb (z) = bz = exp(ln(b)z)
exponential on the base b−1
expcb,λ (z) = Fλ,d (c + Fλ,d(z))
iterated exponential developed at λ(z+t)κ
E(z) = eparameter to expand F (z) in (2.2)η = χ−1
inverse Schröder function; F (z) = η(seκz )ηngeneral Taylor coefficient of the inverse SchröderFsuper-function of base-function h, see (1.1) (2.22)
Fλ,d(C, 0 7→ d) super-function of expb developed at λ; for ex.,Fλprincipal super-function
2
F2,1 0 (0) ≈ 1.69973893
mathematical constantΦ = F2,1 0 (−1) = ln(2)G(E(z)) = F (z)−λdisplaced super-function of modified argument
gn , n ∈ N
Taylor coefficients of function G at zero
Hk,x = {z/k ∈ C : <(z) ≤ x}
range of holomorphizmhbase-function (or the transfer function)hn
for n ∈ N, Taylor coefficients of h at λIinterval,
wherethe regular superfunction is defined by√i = i = −1
imaginary unityj ∈ R, j > 0
used in the example h(z) = z j , F (z) = exp(j z )K = Kλ = h1 = h0 (λ)
derivative of base-function h at λK2 = h0 (2) = ln(2)derivative of h at the fixed point λ = 2
K4 = h0 (4) = 2 ln(2)derivative of h at the fixed point λ = 4
κλ = ln(Kλ )wave-number
κ2 = ln(ln(2)) ≈ −0.36651292058166432701
wave-number for λ = 2
κ4 = ln(2 ln(2)) ≈ 0.32663425997828098238
wave-number for λ = 4
λfixed point, solution of h(λ) = λlnlogarithm, inverse of exponentiation; ln = loge ; exp(ln(z)) = zm, nused as integer variablesN
set of natural numbersR
set of real numbers
Ogrows up not faster (or decays not slower), than its argumentoinfinitezimal parameter, F2,1 (−2.5+oi) refers to upper manifold◦
infix operation of combination of two functionsp, qdependent variables, < and = parts of a function
π = 4 arctan(1) ≈ 3.14
−i/2 of period of exp
rvarious radii
real function : any f such that f (z ∗ ) = f (z)∗ for all z where f is definedconstant factor in the argument of the inverse Schröders = Kts2,1 = V (−1) ≈ −0.63209866105083
Schröder parameters2,3 = V ( 1) ≈ 2.1844747586390
Schröder parameters4,3 = U (−1) ≈ −1.8604041940071
Schröder parameters4,5 = U ( 1) ≈ 0.69470714714324
Schröder parameterσ = exp(−2π 2 /T2 ) ≈ 4·10−24
upper bound for |w(x)| at x ∈ R
T2 = 2πi/κ2 ≈ −17.143148179354847104 i
period of F2,1 and F2,3
T4 = 2πi/κ4 ≈ 19.236149042042854712 i
period of F4,5 and F4,3

ref.
(1.3)

(1.4)
(7.2)
(2.4)
(2.21)
(1.4)
(5.3)
(5.3)
(8.4)
(2.2)
(2.2)
Prop.7(1.1)(2.1)(2.24)

(2.8)
(3.1)
(4.1)
(3.2)
(3.2)
(4.2)

(2.1)

(8.9)
(2.17)
fig.1(3.15)(2.22)(3.10)(5.1)(5.2)(4.10)(5.13)(3.15)(3.15)

30

DMITRII KOUZNETSOV AND HENRYK TRAPPMANN

symboldescription

−1
−1
4πi
Tslog = ln(2)≈ 18.129440567308775239i
period of F2,1
, F2,3, expcb,2tdisplacement of argument of super-function
t2,1 = ln(s2,1 )/κ2 ≈ 1.251551478822188
displacement for F2,1 (0) = 1
t2,3 = ln(s2,3 )/κ2 ≈ 0.880366308272851
displacement for F2,3 (0) = 3
t4,3 = ln(s4,3 )/κ4 ≈ 1.90057764535874
displacement for F4,3 (0) = 3
t4,5 = ln(s4,5 )/κ4 ≈ 1.11520724513161
displacement for F4,3 (0) = 5
θmodification of argument of super-function−1
θ2,4 (z) = F4,3
F2,3 (z) −z ≈ θ̃2,4 (z)θ̃2,4 (z) = ξ · sin(φ+2πz) − sin(φ)
approximation of θ24 (z)ξ = 0.7252175546·10−24 ≈ 0.2 σparameter in approximation θ̃24
φ = 0.74633983476
parameter in approximation θ̃24uinverse Schröder function for λ = 4
un , Un
Taylor coefficients of G and G−1 for λ = 4v
inverse Schröder function for λ = 2
vn , V n
Taylor coefficients of G and G−1 for λ = 2
w(z) = F4,3 (z)−F2,3 (z)deviation between two super-exponentials
ω = E(z)small parameter used to expand function G
x,yused as independent real variableszcomplex variable; sometimes, z = x+iy
Z
set of integer numbers

ref.
(6.7)
(2.4)
(5.3)
(5.4)
(5.5)
(5.6)
(1.3)
(5.10)(5.10)(5.10)(5.10)(4.9)
table 2
(3.9)
table 1
(5.11)
(2.7)
Fig.1
Fig.1

Dmitrii Kouznetsov, Institute for Laser Science, University of Electro-Communications
1-5-1 Chofugaoka, Chofushi, Tokyo, 182-8585, Japan
E-mail address: dima@ils.uec.ac.jp
Henryk Trappmann, Kameruner Str. 9, 13351 Berlin, Germany
E-mail address: henryk@pool.math.tu-berlin.de