Talk:Tetration/Archive 1

message: This page should have some "love" just like the hyper operator page. I have done a lot of work on tetration, and its many connections, so I think I can shed some clarity on this very confusing subject, I'm planning on adding two pages: a page about the Superlogarithm, and a page about Abel's Linearization Equation, does anyone know how to make a new page? Andrew Robbins 23:25, 13 December 2005 (UTC)
replies: text: Its been awhile since I said this, but I'm serious about it this time. AJRobbins (talk) 02:53, 20 November 2007 (UTC)

text: The reason why I want to write a page about the Superlogarithm, is that I found a way to define it over all real numbers, which naturally extends tetration to all real numbers. This method, in the limit produces an infinitely differentiable function, and the method is very simple. My paper describing the method is complete, and I only need to find a place to publish it now. Where do I publish? and can I at least reference my paper in a wikipedia article on the superlogarithm? or do I need to wait until others read my paper and let someone else write the wikipedia article? Andrew 10:23, 21 December 2005 (UTC)
replies: text: I think I remember reading somebody on the internet (was it Daniel Geisler?) who had some doubts about the  superlogarithm extension to real numbers that you mentioned, which goes to show the wisdom of the "no original research" policy.  --Whiteknox 22:13, 28 November 2006 (UTC)

text: How exactly did you define tetration in your 1990 paper? Andrew 06:11, 8 February 2006 (UTC)

text: Take Faà di Bruno's formula and substitute g(x)=f^{\ n-1}(p) where p is a fixed point f(p) = p. This gives the derivatives of an iterated function at a fixed point that leads to the Taylor series of an iterated function in the complex plane. All that is assumed is that f(z) has a derivative and a fixed point not at infinity. The iterated Faà di Bruno's formula works for tetratrion, pentation, hexation and so on.  D^m f^n(p)=\sum_{k=0}^\infty {c_k f'(p)^k + D^m f^{n-1}(p)} which can be solved by noting the difference equation results in a geometrical progression. In the 1990 paper I took the natural step of simplify the geometrical progressions which introduces the terms throughout the results of 1/ 1-f' (p)^m. In my 1990 paper I didn’t treat the case where f' (p)^m = 1 which causes my formula for iterated functions to blow up; for tetration in the real numbers this happens at e^{1/e} and e^{-e}. So I gave a definition for hyperbolic tetration with a fixed point. Daniel Geisler 01:08, 13 February 2006 (UTC)
thread_title: Revision

message: Using the relation n\uparrow\uparrow k = \log_n \left(n\uparrow\uparrow (k+1)\right) (which follows from the definition of tetration), one can derive (or define) values for n\uparrow\uparrow k where k \in {-1, 0, 1}. 
\begin{matrix}
  n\uparrow\uparrow 1
    & = &
  \log_n \left(n\uparrow\uparrow 2\right)
    & = &
  \log_{n} \left(n^n\right)
    & = & 
  n \log_{n} n 
    & = & 
  n
\\
  n\uparrow\uparrow 0
    & = &
  \log_{n} \left(n\uparrow\uparrow 1\right)
    & = & 
  \log_{n} n
    & & & = &
  1
\\
  n\uparrow\uparrow -1
    & = &
  \log_{n} \left(n\uparrow\uparrow 0\right)
    & = &
  \log_{n} 1
    & & & = & 
  0
\end{matrix}
Daniel Geisler
 Tetration just like another mathematics needs to have an axiomatic basis. The trick is to properly enumerate the list of possible consistent systems which extend tetration beyond the positive integers. Negative integers, rational, real and complex numbers are possible examples. 
n\uparrow\uparrow k, k > 0 is computed by iterated exponentiation is standard.'''
n\uparrow\uparrow k, k < 0 can extend k into the negative integers by using iterated logarithms instead of iterated exponentiation, but this means that n\uparrow\uparrow k becomes multivalued.
This confirms the intuitive definition of n\uparrow\uparrow 1 as simply being n. However, no further values can be derived by further iteration in this fashion, as \log_n 0 is undefined.
Articles on arithmetic including treat n\uparrow\uparrow 1 as an axiom, not something capable of confirming an intuition. Furthermore, the dynamics of the Riemann sphere have no problem with dealing with \log_n 0 or the further logarithmic iterations from zero.
Similarly, since \log_{1} 1 is also undefined (\log_{1} 1 = \ln 1{/}\ln 1 = 0/0), the derivation above does not hold when n = 1. Therefore, 1\uparrow\uparrow{-1} must remain an undefined quantity as well. (The figure 1\uparrow\uparrow{0} can safely be defined as 1, however.)
Again, 0^0 is an undefined quantity, so values for 0\uparrow\uparrow{k} cannot be defined directly. However, \lim_{n\rightarrow0} n\uparrow\uparrow{k} is well defined, and exists:
:\lim_{n\rightarrow0} n\uparrow\uparrow k = \begin{cases} 1, & k \mbox{ even} \\ 0, & k \mbox{ odd} \end{cases} 
This limit holds for negative n, as well. 0\uparrow\uparrow{k} could be defined in terms of this limit, but 0\uparrow\uparrow2 = 0 would conflict with the standard undefinedness of 0^0.
I've seen text where 0^0=1 and where 0^0 is undefined. Take a look at the enlarged version of the tetration by period fractal. Zero is in the yellow circle immediately to the left of the large red area. The yellow area is period two, it doesn't converge to a single value but oscillates between two values. The following shows that defining 0^0=1 is consistent with the period two behavior in the neighborhood of zero.
0=0
0^0=1
0^{0^0}=0^1=0
0^{0^{0^0}}=0^0=1
Tetration base  -{10}^{-6},
 1.,-1.\,{10}^{-6},
  1.00001 - 3.14164\,{10}^{-6}\,i ,
  -9.99819\,{10}^{-7} - 
   8.67906\,{10}^{-11}\,i ,
  1.00001 - 3.13987\,{10}^{-6}\,i ,
  -9.99819\,{10}^{-7} - 
   8.67592\,{10}^{-11}\,i ,
  1.00001 - 3.13987\,{10}^{-6}\,i ,
  -9.99819\,{10}^{-7} - 
   8.67592\,{10}^{-11}\,i ,
Tetration base  {10}^{-6},
1.,1.\,{10}^{-6},0.999986,
  1.00019\,{10}^{-6},0.999986,
Daniel Geisler 19:32, 6 May 2005 (UTC)
thread_title: Thoughts on the "Extension to low values of the second operand" 

message: 1. Does anyone know how to define a@b when b is not an integer?? If a is not an integer, it is almost as easy as it is if both are integers. 1.5@2 = 1.5^1.5, but how about if b is not an integer??
> This is an open problem in mathematics. I've seen attempts to define them using fractals, combinatorics, and dynamics, but there's not that much progress.
2. Is e@2 known to be irrational?? The first few decimal places are 15.154262241479.
3. How about e@3, which is around 3814279.1047602??
:I do not think the "subject" has been sufficiently studied to give an answer to those questions. Question 1 may either be senseless or be trivial via logarithms and exponentials. Just my 2c. Pfortuny 19:06, 7 Mar 2004 (UTC)
replies: text:  In one of Rudy Rucker's books, he calls this idea "tetration". -- Tarquin 19:11, 7 Mar 2004 (UTC)
replies: text:  I don't understand your point. I said there are other names for this concept that we should perhaps mention in the article -- Tarquin 14:13, 8 Mar 2004 (UTC)
thread_title: Questions about super-exponentation

message: I would like to create an article about the inverse operation of super-exponentiation, used by the same source that has the @ symbol for super-exponentiation using the & symbol. However, I don't know what to call it, and I feel afraid someone will very likely put it on vfd. Do you know?? 66.32.82.95 17:56, 2 Apr 2004 (UTC)
thread_title:  Inverse of super-exponentiation 

message: True or false: there are plenty of Wikipedia links that change in the following category:
Originally, the link at Article A was a direct link to Article C, but later, someone modifies it and makes Article A link to Article B where Article B re-directs to Article C.
There appear to be plenty in the case of this article being C, but I want to know if there are plenty with no particular Article C. 66.245.23.108 22:56, 12 Jul 2004 (UTC)
thread_title:  True or false?? 

message: Can anyone think of a category for this article to go into?? 66.245.77.90 00:36, 26 Aug 2004 (UTC)
thread_title:  Categorizing this article 

message: I've placed soft hyphens (­) into the 155-digit (206-character) value of 4↑↑3 because it was making the page super-wide in my browser. - dcljr 05:08, 29 Aug 2004 (UTC)
thread_title:  Soft hyphens in long numbers 

message: The last three entries on the page read:
:n↑↑(-1) = 0 for all real numbers not equal to 1
:n↑↑(-2) = negative infinity for all real numbers greater than 1
:n↑↑(-2) = infinity for all real numbers between 0 and 1
Are these by definition? (Whose?) Can someone explain to me (and in the article itself) what a negative "super-exponent" (or whatever you'd call it) would even mean? - dcljr 05:22, 29 Aug 2004 (UTC)
replies: text: What would make more sense to me -- and probably to dcljr -- is what I've replaced that list of identities with. Even if my TeX skills leave much to be desired. --Aponar Kestrel  (talk) 20:24, 2004 Sep 15 (UTC)
But ln (x) approaches –∞ as x approaches 0. --116.14.26.124 (talk) 10:31, 23 June 2009 (UTC)
thread_title:  Negative "super-exponents"? 

message: On July 4, 2004, User:Gdr asked this page to be renamed, but it hasn't been renamed. What happened?? 66.245.71.98 15:45, 16 Sep 2004 (UTC)
replies: text: Your helpful wizards have finally woken up and noticed! Noel 20:01, 17 Sep 2004 (UTC)
thread_title:  Page title 

message: Given that, as pointed out in the article, the up-arrow symbol (when present on a computer keyboard) is used similarly to the caret symbol (indicating superscripts), couldn't we just replace all the double up-arrows on the page (except the ones next to Knuth's name) to double carets (^^)? This would make the page readable in all browsers. Just a suggestion... - dcljr 06:56, 10 Oct 2004 (UTC)
thread_title:  Carets 

message: Please read very slowly and carefully:
Can anyone come up with a way to name numbers that tetration can help visualize the magnitude of?? One way that I came up with is a building with 103 floors, each of which is for numbers of various sizes; the higher the floor, the larger the numbers. The Tetrational System (the numbering system that this uses) has the following number names:
For numbers <= 10^30, whose magnitudes are easy to visualize without tetration, it uses the same number names as Rowlett. On the first 2 floors, we have:
First floor: Two, Three, Six, Nine, Twelve, Fifteen, Eighteen, Twenty-one, Twenty-four, Twenty-seven, Thirty
Second floor: Hundred, Thousand, Million, Gillion, Tetrillion, Pentillion, Hexillion, Heptillion, Oktillion, Ennillion, Dekillion
Now, let's go onto the third floor. This is where the numbers get large enough that tetration is a useful way; each term is 10 to the power of the previous term:
Third floor: Googol, Froogol; the remaining words on this floor are the same as those on the second floor only that they use -illoogol instead of -illion (that is, Milloogol to Dekilloogol are 10^(10^6) through 10^(10^30.) (Froogol is a back-formation on Froogle on the model of Googol/Google.)
Fourth throught 103rd floors: Simply take the number names on the third floor and add "-plex" for those on the fourth floor, "-duplex" for the fifth floor, "-triplex" for the sixth floor, and so on all the way to "-centuplex" for the 103rd floor. This makes the largest number in the building (dekilloogolcentuplex) 10^10^10^...10^10^10^30 with a total of 103 numbers (102 tens and a 30) between exponent signs.
Any numbers too large for this?? At this moment, I know of only one number too large for this building, Graham's number. 66.245.98.219 23:12, 16 Nov 2004 (UTC)
thread_title:  Tetration number names 

message: The article seems to suggest that tetration is the logical extension of the sequence of addition, multiplication, and exponentiation.
However, what about ((nn)n...) for m occurences of n?
It seems to me this operation is equally an extension: Since exponentiation is non-commutative, the sequence bifurcates at this point.
This "inferior super-exponentiation", if you will, has interesting properties of its own.
For instance, a negative value for m corresponds to taking the nth root m times (since it can remove exponents), so m=0 yields \sqrt[n]n.
Anyway, shouldn't there be an article about this operation also?
Who here knows its standard name(s) and notation(s), or does it have any?
(I thought "iterated exponentiation" was reasonable, but the article's (and discussion's) usage conflicts.)
--Ddawson 12:08, 20 Mar 2005 (UTC)
replies: text: This is simply nnm n^{n^m}(NESTED SUBS/SUPS ARE BROKEN ON WIKIPEDIA >:()  , there is nothing especially "new" about it relative to the sense that tetration is some new operation. Dysprosia 22:25, 20 Mar 2005 (UTC)
thread_title:  The other successor of exponentiation 

message: Someone made a strange removal on the basis that "2 is greater than e" (??). I have reverted that, but then I noticed that the reason that the article stipulates r <= e, is to ensure "x is not more than e^{1/e}". But investigation shows that r^(1/r) is never greater than e^(1/e). The article misleadingly implies otherwise when r > e. Therefore I think that that phrase should be removed or revised. Eric119 02:28, 2 August 2005 (UTC)
replies: text: I rephrased it.--Patrick 07:40, 2 August 2005 (UTC)
thread_title: Infinite power towers

message: {{a + b = a + } \atop {\ }} {{\underbrace{1 + \cdots + 1}} \atop b}
Or, a incremented b times. Or, applying b times the Successor function to a (See Peano axioms, it's all there)
Locoluis̊ 01:06, 31 March 2006 (UTC)
replies: text: Just my tought. If tetration is "level 4", exponentiation is "level 3". multiplication is "level 2" and addition is "level 1" when the addition with 1 is "level 0" and "ground level". //Fabben
----
I have been trying to find a solution of "level 0 binary operator" for a few years.
The hypothesis of increment by 1 proposed above, looks quite reasonable, however it leaves
some blank spots in the whole picture.
Let's accept that a(1)b = a+b, a(2)b = ab, a(3)b = a^b etc. (just for further convenience).
Now we are discussing the problem of a(0)b. As Locoluis proposes, a(0)b = a+1. There are two logical conclusions from that:
1) The "level 0" operator is not commutative: a(0)b = a+1 != b(0)a = b+1
2) The function f(x)=a(0)x is a constant, and it does not depend on its "x" argument.
From the other hand, let's examine the following sequence:
a(1)a = a(2)2, i.e. a+a = a*2;
a(2)a = a(3)2, i.e. a*a = a^2;
a(3)a = a(4)2, i.e. a^a = a^^2 etc.
and let's try to extrapolate that sequence below "level 1". We will get:
a(0)a = a(1)2, i.e. a(0)a = a+2,
which comes into a contradiction with the assumption that a(0)b = a+1 (!)
In my opinion, the a(0)b operator definition should meet the following requirements:
1) a(0)a = a+2 (see explanation above);
2) a(0)m < a(0)n, if m<n;
3) m(0)a < n(0)a, if m<n;
4) a < a(0)b < a+b for any positive integer "a" and "b".
A very special case is when b=0. Accepting our assumption, we get
0(0)0 = 0+2 = 2, which is larger than 0(1)0 = 0+0 = 0 (!!!)
--Beloturkin 09:30, 8 September 2006 (UTC)
thread_title:  A thought about addition 

message: Isn't this page violating the third paragraph of "What Wikipedia is not"?
:This should be added as a comment below.  Apparently this is refering to the section on original research ([Wikipedia is not, 1.3]).  I personally think it can be deleted because nobody signed to it.  --Whiteknox 19:20, 28 December 2006 (UTC)
thread_title:  Moved Comment 

message: Would it help in extending tetration to the reals to extend it first to n^^(1/2) (and by extension n^^(k+1/2))?  If so, what useful identities can be used to solve the equation x = n^^(1/2) for n?  At first I thought about saying x = n^^(1/2) implies n = x^^2, but this really doesn't make sense, since (n^^a)^^b doesn't equal n^^(a*b) or n^^(a+b) or anything else simple.  ((n^^2)^^2 = n^(n^(n+1)) = (n^^3)^n.)  
Using the symbol "=/=" to denote "not equal to";
It sounds OK, the problem is though that in general n^^(1/2) =/= n^^(2/4), n^^(3/6), n^^(4/8) etc. which would be necessary for this extension to be rigorous (and mathematics is always rigorous). Other oddities crop up as well, such as the fact that 2^^(1/3) < 2^^(2/7) even though 1/3 > 2/7.
This and other problems in extending tetration to the reals is a consequence of the fact that exponentiation is neither commutative nor associative, so that in general a^b =/= b^a and a^(b^c) =/= (a^b)^c.  
Meltingpot (talk) 09:19, 15 April 2008 (UTC)
thread_title:  n ^^ 1/2 

message: Has this special case got a name?  It is barely mentioned either here or on Exponentiation.  —DIV (128.250.80.15 (talk) 07:43, 1 May 2008 (UTC)
thread_title:  Name? 

message: Not my area of expertise, but the following two sentences in this section appear to be in direct conflict:
    Extending x \uparrow\uparrow b to real numbers x > 0 is straightforward...
    At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex numbers...
Can someone clarify this?
Ultraviolet777 22:58, 24 July 2007 (UTC)
replies: text: The latter bit is talking about real or complex values of "b", the number of iterated exponents.  I changed the wording a little in the article to improve this.  Doctormatt 23:18, 24 July 2007 (UTC)

text: It appears that you haven't carefully read Doctormatt's changes or comments.  ^b a is analytic in a for fixed (integer) b.  He removed (IMHO) any indication that there is a simple extension to real b of any sort, even if uxp is offered as such an extension.  — Arthur Rubin |  (talk) 07:46, 21 January 2008 (UTC)
replies: text: Dear Artur. Sorry if my typing above is not clever enough. I understand that at fixed integer b the tetration is analytic function of a. Doctormatt typed about that case. Perhaps, his typing cannot be applied to the case of real b. The extension to the real heighs has two examples. First of them seems to be just uxp. The second one has smooth first derrivative, but it is still not analytic, and the series above in not valid. May I repeat the question: How about the analytic extension for real and for complex b? I suggest that somebody gives the reference to such an extension; overwice we should type that <no such extension is yet reported>. dima (talk) 09:14, 21 January 2008 (UTC)
replies: text: A paper describing a real-analytic solution for the infra log (base e) was reported to me in a USENet post in the 90s, and (I believe) here in Wikipedia:Reference desk/Mathematics.  I never looked up the paper, so I can't confirm it.  — Arthur Rubin |  (talk) 13:54, 21 January 2008 (UTC)
thread_title:  Real Extension 

message: Well, Doctormatt, on the base of your update of the "real extension", could you please, name, for example, the first terms of the expansion of the tetration in the Tailor series at samall values of the number ~b~ of exponentiations,
? 
In other words, if
 ~{^b\rm e}=1+\sum_{n=1}^{\infty} c_n b^n 
then, what are coefficients ~c~?
dima (talk) 02:24, 21 January 2008 (UTC)
replies: text: It appears that you haven't carefully read Doctormatt's changes or comments.  ^b a is analytic in a for fixed (integer) b.  He removed (IMHO) any indication that there is a simple extension to real b of any sort, even if uxp is offered as such an extension.  — Arthur Rubin |  (talk) 07:46, 21 January 2008 (UTC)
replies: text: Dear Artur. Sorry if my typing above is not clever enough. I understand that at fixed integer b the tetration is analytic function of a. Doctormatt typed about that case. Perhaps, his typing cannot be applied to the case of real b. The extension to the real heighs has two examples. First of them seems to be just uxp. The second one has smooth first derrivative, but it is still not analytic, and the series above in not valid. May I repeat the question: How about the analytic extension for real and for complex b? I suggest that somebody gives the reference to such an extension; overwice we should type that <no such extension is yet reported>. dima (talk) 09:14, 21 January 2008 (UTC)
replies: text: A paper describing a real-analytic solution for the infra log (base e) was reported to me in a USENet post in the 90s, and (I believe) here in Wikipedia:Reference desk/Mathematics.  I never looked up the paper, so I can't confirm it.  — Arthur Rubin |  (talk) 13:54, 21 January 2008 (UTC)
thread_title: real <math>~b~</math>

message: I'd like to excuse for doing edits without being logged in (usually I automatically logged in), so there is no author at my (small) series of edits just before.
Don't remember, how to correct this - sorry
Gottfried Helms
Gotti 08:22, 2 November 2007 (UTC)
thread_title:  some small edits without author-notification  

message: Why not bother to add a Pentation article? Even though it's a redirect, I think it should have its own article. Pilover819 (talk) 11:37, 23 December 2007 (UTC)
replies: text: Actually, it would be nice to have a page on pentation, as there is at least one result that has been found about it. For example Jay D. Fox notes in the Tetration Forum that the limit \lim_{n\rightarrow (-\infty)} x {\uparrow}{\uparrow}{\uparrow} n is the lower fixed point of real-valued tetration, for example: e{\uparrow}{\uparrow}(-1.85) = -1.85 so e {\uparrow}{\uparrow}{\uparrow} (-\infty) = -1.85. Aside from this result, though, not much is known about pentation. AJRobbins (talk) 16:08, 28 December 2007 (UTC)
thread_title: Pentation

message: I am not satisfied with the paper. The definition of tetration is almost absent. The speculation about a^{a^{...^a}}_{^{~x {\rm times}}} looks as property of ~ ^xa ~ in the specific case of real ~a~ and natural ~x~. The definition should allow to see that this function is analytic with respect to each of its arguments, (at least in some range of values of arguments), reveal its singularities, and so on. Overwice we should write at the preamble that   Tetration is some mathematical function of two variables which is believed to be definable as analytical in some (still unknown) set including natural numbers. P.S. Look at the definitions of other special functions; try to do the same for tetration. Then we can keep tetration among special funcitons. dima (talk) 04:39, 13 January 2008 (UTC)
replies: text: I have since come to an understanding that (Claim E) has a major typo in it (should be (-1)+ not 0+, which makes a big difference, as noted by Arthur Rubin). This means that the extension called "UXP" is nothing more than the method described in the "linear approximation" section of this article. This indicates that a new article ultra exponential function is unnecessary. I also agree with most of the comments in the WikiProject Mathematics discussion about "UXP", and that the "UXP" article uses non-standard terminology with non-standard notation, to introduce a standard extension of tetration in a way that makes it seem different when it really isn't. AJRobbins (talk) 02:25, 20 January 2008 (UTC)

text: This could have been avoided if  Dima and Ultra had found the Tetration Forum before Wikipedia. The purpose of the forum was to have a place to talk about terminology, notation, and methods so that things like this don't happen. The Tetration Forum is also host to many discussions about series expansions, and extensions to the complex plane. AJRobbins (talk) 02:25, 20 January 2008 (UTC)

text: I see, some users dislike the theorem above, but please, will you return to the article at least the reference? dima (talk) 03:26, 24 January 2008 (UTC)
replies: text: I consider the theorem uninteresting, thereby making the paper uninteresting.  Others (including the author) may disagree....  — Arthur Rubin |  (talk) 03:40, 24 January 2008 (UTC)
thread_title: Dubious

message: I have moved this section from the main article, as its claims are unclear, and if I interpret them correctly, contradictory. You cannot have differentiability at one integer value (Claim C), and have non-differentiability at another integer value (Claim E) while you have a functional equation that relates these two values (Claim A). That is simply impossible. If anyone can explain to me how these claims are not in contradiction with themselves, please do so. As for you Domitori/dima and Ultra.Power, there is an excelent forum for discussing these things, called the Tetration Forum. I encourage both of you to join and learn more about tetration, and also discuss this ultra-exponential-function you are writing about. It is a great place to discuss tetration so that you can get feedback on your theories and gaps in your knowledge. AJRobbins (talk) 06:43, 19 January 2008 (UTC)
replies: text: I have since come to an understanding that (Claim E) has a major typo in it (should be (-1)+ not 0+, which makes a big difference, as noted by Arthur Rubin). This means that the extension called "UXP" is nothing more than the method described in the "linear approximation" section of this article. This indicates that a new article ultra exponential function is unnecessary. I also agree with most of the comments in the WikiProject Mathematics discussion about "UXP", and that the "UXP" article uses non-standard terminology with non-standard notation, to introduce a standard extension of tetration in a way that makes it seem different when it really isn't. AJRobbins (talk) 02:25, 20 January 2008 (UTC)

text: This could have been avoided if  Dima and Ultra had found the Tetration Forum before Wikipedia. The purpose of the forum was to have a place to talk about terminology, notation, and methods so that things like this don't happen. The Tetration Forum is also host to many discussions about series expansions, and extensions to the complex plane. AJRobbins (talk) 02:25, 20 January 2008 (UTC)
thread_title:  Discussion about "UXP" 

message: The functional approach to the tetration leads to so-called ultra exponential function
~{\rm f}(b)={\rm uxp}_a(b)~, determined with following condiitons:
 (A) ~{\rm f}(x)=a^{{\rm f}(x-1)}~~ \mbox{for all}~~~ x>-1 ~
 (B) ~{\rm f}(0)=1 
 (C) ~\rm f~ is differentiable on ~(-1,0)~
 (D) ~{\rm f}'~ is non-increasing or non-decreasing at ~ (-1,0)~
 (E) \lim_{x\rightarrow 0^+}{\rm f}'(x)=\ln(a) \lim_{x\rightarrow0^-} {\rm f}'(x),
It happens that for real values of ~a~, the conditions (A-E) unambiguously determine the funciton ~{\rm uxp}_a~ 
, 
and, for integer values of the argument, the ~ {\rm uxp}~ function coincides with tetration defined above
. 
See the specific article Ultra exponential function for the details.
At ~a \ne {\rm e}~ the function ~{\rm uxp}_a~ is not smooth, and the alternative approaches to the extension of the tetration exist, which lead to the smooth functions.
:I see, some users dislike the theorem above, but please, will you return to the article at least the reference? dima (talk) 03:26, 24 January 2008 (UTC)
replies: text: I consider the theorem uninteresting, thereby making the paper uninteresting.  Others (including the author) may disagree....  — Arthur Rubin |  (talk) 03:40, 24 January 2008 (UTC)
thread_title:  Ultra exponential function 

message: I discovered this pattern to do with x^x the other day and I thought it was quite interesting.
:\frac {dx^x} {dx} = x^x (\ln x + 1) 
:\frac {d^2x^x} {dx^2} = x^x (\ln^2 x + 2\ln x + \frac {x+1} {x}) 
:\frac {d^3x^x} {dx^3} = x^x (\ln^3 x + 3\ln^2 x + 3\frac {x+1} {x}\ln x + \frac {x^2+3x-1} {x^2}) 
:\frac {d^4x^x} {dx^4} = x^x (\ln^4 x + 4\ln^3 x + 6\frac {x+1} {x}\ln^2 x + 4\frac {x^2+3x-1} {x^2}\ln x+ \frac {x^3+6x-x+2} {x^3}) 
The nth derivative takes a similar form, with a pattern taken from Pascal's triangle and another pattern of the polynomials/(their degree) continuing down in the same positions relative to the greatest power of lnx.  Lol, the only mildly tedious thing was proving the conjecture I made that this pattern was true for all natural n.  If anyone has seen these polynomials before, please let me know.  I'd like to know if they have a specific name.  If you need to know any more of them, I wrote a program that generated the first 172 of them, before the floating point numbers stopped working.--Steven Weston (talk) 23:25, 6 July 2008 (UTC)
thread_title:  Interesting pattern 

message: As proposed earlier at Talk:ultra exponential function#This is nothing new., I'm proposing that ultra exponential function be merged into Tetration#Extension to real heights.  — Arthur Rubin  (talk) 02:58, 17 August 2008 (UTC)
replies: text: Agree.--Dojarca (talk) 15:20, 31 August 2008 (UTC)

text: Agreed AJRobbins (talk) 02:32, 17 September 2008 (UTC)

text: Agree. May be, even Tetration#Piece-vice extension to real heights. dima (talk) 08:41, 27 September 2008 (UTC)
thread_title:  Ultra exponential 

message: I am speculating about another criteria, which may make fractional iteration unique. Assume b = sqrt(2), t2=2, t4=4 such that b = t2^(1/t2) = t4^(1/t4) . Assume also Tb(x) = b^x and Tb(x,h) the h'th iterate and also the "decremented tetration" Ut(x) = t^x - 1 and Ut(x,h) the h'th iterate. 
Then powertowers T_b(x,h) of b can be expressed by "decremented powertowers" 
   T_b(x,h) = (U_t2(x/t2-1,h)+1)*t2 = (U_t4(x/t4-1,h)+1)*t4 
at least for integer heights. This construction relates tetrational functions of different bases with each other and introduces then new requirements for the fractional iteration as well. Although this restriction occurs yet only for "decremented" tetration, I think this idea has some importance and may be thought further... —Preceding unsigned comment added by Druseltal2005 (talk • contribs) 
Gotti 17:50, 13 September 2008 (UTC)
replies: text: I don't think this is correct.  Do you wish to further comment?  — Arthur Rubin  (talk) 21:34, 9 November 2008 (UTC)

text: Yes, the formula seems incorrect, as well as not being obviously related to the tetration concept(s) described here.  — Arthur Rubin  (talk) 06:50, 6 February 2009 (UTC)

text: Seems to work for b < e1/e, where there are real fixed points available for T, and hence U — although the fact that t=2 and t=4 seem to produce different "natural" values for fractional iterates suggests that this problem is even more difficult than it appears....  — Arthur Rubin  (talk) 22:33, 14 February 2009 (UTC)
thread_title: Decremented tetration

message: I would like some feedback on whether adequate sources can be found to justify having a section of the article about the following interesting fact: 
Given any positive integers b and d, all towers b\uparrow\uparrown with integer n > d + 1 have the same d rightmost decimal digits.  The following is an extremely simple algorithm that produces these digits, assuming b is not divisible by ten (if b is divisible by ten, then the d digits are all 0):
 Let x0 = b, and compute xi = bxi - 1 mod 10d (i = 1, 2, 3, ...), stopping when xi = xi - 1 mod 10d. 
When the algorithm stops, xi will be the required d-digit string (in base-10, omitting any leading 0s), and the value of i will be the height of the shortest "base b" tower that has these d rightmost digits. 
E.g., here are the ten rightmost digits of each sufficiently tall tower with b = 1,2,...,10, respectively — a tower is sufficiently tall if its height is equal to or greater than the indicated height h:
 b   b^^x, x ≥ h    h
 --  -------------  --
 1      0000000001  1   (with leading 0s inserted) 
 2   ...3432948736  12
 3   ...2464195387  11
 4   ...0411728896  11
 5   ...8408203125  4
 6   ...7447238656  9
 7   ...1565172343  6
 8   ...6895225856  11
 9   ...7392745289  10
 10  ...0000000000  2
The algorithm and the "interesting fact" in italics above are described in (or, rather, I deduced them from) the discussion on this page. I'm not sure whether that's an adequate source to justify putting these conclusions into the article, though. --r.e.s. (talk) 08:02, 13 September 2008 (UTC)
replies: text: That's certainly not an adequate source.  However, if there were a published article, it could be included, I would think.  — Arthur Rubin  (talk) 21:44, 9 November 2008 (UTC)
thread_title: Decimal digits of power towers

message: Can CX, XC, or XX be differentiated (or anti-differentiated) and if so how? srn347 —Preceding unsigned comment added by 68.7.25.121 (talk • contribs) 00:17, November 6, 2008
:I don't see it.  All I can see that you can get is by formal differentiation of the functional equation.  — Arthur Rubin  (talk) 21:40, 9 November 2008 (UTC)
replies: text:  For C=e , the evaluation of F(X)=XC  and its derivatives is described in reference cited in the article, http://www.ams.org/mcom/0000-000-00/S0025-5718-09-02188-7/home.html  If can be generalized for  C>1. dima (talk) 15:37, 15 February 2009 (UTC)
thread_title:  Derivatives 

message: Hi.
I saw this:
"The term super-exponentiation is the most proper candidate for a name, however, Bromer published his paper Superexponentiation in 1987, and Goodstein published his paper Transfinite Ordinals in Recursive Number Theory (which coined the term tetration) in 1947, which predates Bromer. So although this is not a misnomer, the shorter and older term has gained more use."
But how is that the most proper candidate? "Super-exponential" could just refer to anything that grows faster than exponential. So it is not specific. It could be a double-exponential. It could be pentation. It could be the tetrasquare (x^x). It could even be the generalized factorial function GAMMA(x). "Tetration", and "tetrational", however, refer only to one thing. mike4ty4 (talk) 09:15, 25 January 2009 (UTC)
thread_title:  Naming?