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 Exact and Unique solution for base e^(1/e) jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/11/2007, 10:21 PM (This post was last modified: 08/11/2007, 10:22 PM by jaydfox.) bo198214 Wrote:Quote:But I'm only looking at the first 15 or so terms of the series. It seems pretty well behaved and very well-defined, but maybe I'm missing something? What exactly did you try? Did you experiment with the iterations of $e^x-1$? *wondering*Indeed I did. I found that if you take the first several iterations, and look strictly at the a_n's for a given n, they obey a very well-defined (n-1)th-order polynomial: $\begin{eqnarray} f(x) & = & e^x-1 \\ f^{\circ 0}(x) & = & 0\left(\frac{1}{0!}\right)\ +\ 1\left(\frac{x}{1!}\right)\ +\ 0\left(\frac{x^2}{2!}\right)\ +\ 0\left(\frac{x^3}{3!}\right)\ +\ 0\left(\frac{x^4}{4!}\right)\ +\ \dots \\ f^{\circ 1}(x) & = & 0\left(\frac{1}{0!}\right)\ +\ 1\left(\frac{x}{1!}\right)\ +\ 1\left(\frac{x^2}{2!}\right)\ +\ 1\left(\frac{x^3}{3!}\right)\ +\ 1\left(\frac{x^4}{4!}\right)\ +\ \dots \\ f^{\circ 2}(x) & = & 0\left(\frac{1}{0!}\right)\ +\ 1\left(\frac{x}{1!}\right)\ +\ 2\left(\frac{x^2}{2!}\right)\ +\ 5\left(\frac{x^3}{3!}\right)\ +\ 15\left(\frac{x^4}{4!}\right)\ +\ \dots \\ \end{eqnarray}$ It's easier to make out the patterns when the coefficients are laid out in a matrix, without all the other stuff to confuse things: $\begin{array}{c|ccc} & 0 && 1 && 2 && 3 && 4 && 5 && 6 && 7 & \dots\\ \hline 0 & 0 && 1 && 0 && 0 && 0 && 0 && 0 && 0 & \dots\\ 1 & 0 && 1 && 1 && 1 && 1 && 1 && 1 && 1 & \dots\\ 2 & 0 && 1 && 2 && 5 && 15 && 52 && 203 && 877 & \dots\\ 3 & 0 && 1 && 3 && 12 && 60 && 358 && 2471 && 19302 & \dots\\ 4 & 0 && 1 && 4 && 22 && 154 && 1304 && 12915 && 146115 & \dots\\ 5 & 0 && 1 && 5 && 35 && 315 && 3455 && 44590 && 660665 & \dots\\ 6 & 0 && 1 && 6 && 51 && 561 && 7556 && 120196 && 2201856 & \dots\\ \vdots & \vdots && \vdots && \vdots && \vdots && \vdots && \vdots && \vdots && \vdots & \ddots \end{array}$ If you try to find an equation for the mth iteration, looking only at the terms in the mth row, you'll be a sad camper. But if you look at any particular column n, you should easily be able to verify that it's terms are part of a sequence defined by an (n-1)th order polynomial. Armed with this knowledge, it should be trivial to calculate the first 15-20 constants for the iteration m=0.5, for example. With a sufficiently powerful math library, one should be able to derive any desired number of terms in the sequence. On a sidenote, I'm using Microsoft Excel with the XNumbers library, but Mathematica or Maple, etc., would be preferable. Alas, I'm not a student (self-study apparently doesn't count), and I can't afford the non-student version. Anyway, the polynomials themselves should behave very well. As such, the m=0.5 iteration should converge much faster than the m=3 iteration, for example, at least in the vicinity of 0. The radius of convergence should be greater than 0, and regardless of how small it might be, any radius of convergence greater than 0 guarantees that analytic extension is possible. However, I've only looked at the first dozen or so polynomials (for the first dozen or so terms of the sequences). There seems to be a well-defined formula for determining the coefficient of the highest degree term of the [/i]n[/i]th polynomial, so it should be possible to prove convergence. I haven't "proven" convergence per se, but I can see the foundation for a proof in the making. andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 08/13/2007, 03:20 PM (This post was last modified: 08/13/2007, 03:21 PM by andydude.) Oh! Now I get it, you showed how to construct the continuous iterate of the base-$e^{1/e}$ logarithm. I see now. Good Job jaydfox, since it's a continuous iterate based on series expansion it must be related to the $e^x - 1$ method, and as such it is unique. Andrew Robbins Gottfried Ultimate Fellow Posts: 758 Threads: 117 Joined: Aug 2007 08/13/2007, 03:45 PM (This post was last modified: 08/13/2007, 06:18 PM by Gottfried.) Hi, these coefficients occur exactly in the powers of the matrix of Stirling numbers 2'nd kind, as I described it in my posting about iteration for e^x-1 and which must be used in a factorial scaled version to obtain the results. Your coefficients in rows for a certain iteration are the coefficients of the second column of the related power of St2 (second column only will be used for the scalar result in my notation). Code:VE(St2,8,6):  \\ extraction of 8 rows, 6 columns   1   .    .     .     .     .   ...   0   1    .     .     .     .   0   1    1     .     .     .   0   1    3     1     .     .   0   1    7     6     1     .   0   1   15    25    10     1   0   1   31    90    65    15   0   1   63   301   350   140   ...   ...                            ... second iteration = second power Code:VE(St2^2,8,6):   1     .      .      .      .     .   ...   0     1      .      .      .     .   0     2      1      .      .     .   0     5      6      1      .     .   0    15     32     12      1     .   0    52    175    110     20     1   0   203   1012    945    280    30   0   877   6230   8092   3465   595   ...   ...                            ... third iteration = third power Code:VE(St2^3,8,6):   1       .       .       .       .      .   0       1       .       .       .      .   0       3       1       .       .      .   0      12       9       1       .      .   0      60      75      18       1      .   0     358     660     255      30      1   0    2471    6288    3465     645     45   0   19302   65051   47838   12495   1365   ...                            ... P.s. If you need a free numb-theoretical package with arbitrary precision you might consider Pari/GP I've also written a convenient user-interface for this usable in win-xp ("Pari-TTY") Gottfried Gottfried Helms, Kassel jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/13/2007, 06:54 PM Wow, that will make life so much easier. I didn't have a formula for calculating the coefficients of the nth iterate, so I had to calculate them by hand (at least until I had enough terms to resolve the polynomial). I think this helps me understand the matrix method better, and why, with an appropriate set of matrices and matrix operations, the two methods should be identical. ~ Jay Daniel Fox jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 07/31/2009, 08:11 PM Well, I've been off with projects (experimenting with random number generators, learning lots about primes, 2-adic numbers, galois fields, spectral tests, lattice reduction, lattice covering and packing, and the like). But I've decided to turn my interests back to tetration and related subjects. One particular subject that I stopped investigating was my "cheta" function, which is essentially a scaled version of the continuous iteration of $\exp(x)-1$. (I stopped work on it mainly because I was using it to find a general solution to tetration using a change of base formula, and I unfortunately found that the results don't match those of the "natural" solution.) So I've started tinkering with it again, slowly remembering old insights and forming new ones. Before I invest too much time, I was wondering how much has been discussed on this subject in my 1-2 year absence. Can someone point me to anything new that I should read up on before I get too far along? Relevant subjects include not only base $e^{1/e}$, a.k.a. eta, but also the problem of non-convergence for non-integer iterates of functions, etc. ~ Jay Daniel Fox andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 07/31/2009, 11:09 PM Well, hey! Welcome back!! Some of us are writing an overview paper of the 5 or more methods we have discovered/developed for real analytic tetration. You might want to see where your cheta function would fit into such a paper. I think it would be a welcome addition (if it is possibly real analytic). Also, I was poised to write the "intuitive" section; a while back Henryk and I decided that "natural tetration" is too easily confused with "base-e tetration", so we use the term "intuitive" now to refer to the method using the Abel matrix (which I think also includes solving with pure power series and the log(z)/log© + powerseries method you discovered). I think both of these would be considered part of "intuitive tetration", and as such should be in that section of the paper. Glad to hear your interest has been rekindled! Andrew Robbins sheldonison Long Time Fellow Posts: 631 Threads: 22 Joined: Oct 2008 07/31/2009, 11:18 PM (07/31/2009, 08:11 PM)jaydfox Wrote: Well, I've been off with projects (experimenting with random number generators, learning lots about primes, 2-adic numbers, galois fields, spectral tests, lattice reduction, lattice covering and packing, and the like). But I've decided to turn my interests back to tetration and related subjects. One particular subject that I stopped investigating was my "cheta" function, which is essentially a scaled version of the continuous iteration of $\exp(x)-1$. (I stopped work on it mainly because I was using it to find a general solution to tetration using a change of base formula, and I unfortunately found that the results don't match those of the "natural" solution.) So I've started tinkering with it again, slowly remembering old insights and forming new ones. Before I invest too much time, I was wondering how much has been discussed on this subject in my 1-2 year absence. Can someone point me to anything new that I should read up on before I get too far along? Relevant subjects include not only base $e^{1/e}$, a.k.a. eta, but also the problem of non-convergence for non-integer iterates of functions, etc.Jay, I become interested in tetration last fall, long after you left this forum, but followed many of the same lines of reasoning that you did, often without realizing it. I eventually went through every one of your posts, and thoroughly enjoyed reading them. I realized you had already explored almost every idea I had, and many more. My math education is somewhat lacking (25 years ago I got an BS in computer engineering), but it has been fun to read about the progress made in tetration. Dimitrii Kouznetsov has some graphs published for $\text{sexp}_\eta$, at http://en.citizendium.org/wiki/Tetration . There was also a lot of interesting posts about the upper super exponential, which I found very interesting, http://math.eretrandre.org/tetrationforu...260&page=2 Henryk and Dimitrii coauthored a paper on that. And there's my own thread, http://math.eretrandre.org/tetrationforu...hp?tid=236, where I explored some of the same base conversion ideas you had earlier explored. I stopped, after realizing that the base conversion definition of the super exponential is probably not analytic, although I don't have the math background to give it a rigorous treatment. I haven't actually seen a published graph of the upper super exponential for base e^(1/e). I personally think its a very interesting base to work with. I don't have the sophisticated math software to make the nice complex contour graphs. - Sheldon Levenstein bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 08/01/2009, 08:21 AM (07/31/2009, 08:11 PM)jaydfox Wrote: But I've decided to turn my interests back to tetration and related subjects. Great. Quote:Can someone point me to anything new that I should read up on before I get too far along? Relevant subjects include not only base $e^{1/e}$, a.k.a. eta, but also the problem of non-convergence for non-integer iterates of functions, etc. We have a uniqueness criterion now for the real-analytic iteration at a conjugated fixed point pair, which is however not so easy to verify. Its about injectivity and imaginary unboundedness of the Abel function on an initial region in the complex plane. The construction of Kneser of the arcsuper-exponential - formerly super-logarithm - (base e, or generalized to >e^(1/e)) satisfies this criterion, nothing is known about the other approaches. Here is the thread of discussion of this paper. Though the proof is from my hand I still doubt if it is correct, perhaps have a look yourself. (07/31/2009, 11:09 PM)andydude Wrote: Some of us are writing an overview paper of the 5 or more methods we have discovered/developed for real analytic tetration. You might want to see where your cheta function would fit into such a paper. Yes, I absolutely second that, there is even a section about your method already in the paper, which I originally filled myself extracting your posts, however my computer crashed and I lost most of my data files. So I restarted that article again, but until now the section is empty. So if you want to participate ... I am also glad to see Sheldon now together with you on this forum as he has perhaps a more intuitive understanding of your ideas than I have. When you mention that it is a "scaled version" of the regular iteration of exp(x)-1, perhaps you should read an article of Walker, which also based an iteration of exp(x) on the regular iteration of exp(x)-1: Walker, P. (1991). Infinitely differentiable generalized logarithmic and exponential functions. Math. Comput., 57(196), 723–733. In this article he also features the by Andrew rediscovered intuitive iteration (we renamed "natural" to "intuitive" as it causes less confusion). I think he uses also his following articles Walker, P. L. (1990). The exponential of iteration of $e\sp x-1$. Proc. Am. Math. Soc., 110(3), 611–620. Walker, P. L. (1991). On the solutions of an Abelian functional equation. J. Math. Anal. Appl., 155(1), 93–110. (here he describes the iteration of b^x for b=e^(1/e) ) I have all these articles as pdfs, so if someone does not have access to suitable libraries, I can send it via private e-mail. For the theoretical properties of the iteration of functions at a parabolic fixed point (f'(p)=1, e.g. f(x)=e^(x/e)) see also the publicly available article "Milnor, dynamics in one complex variable", which I already featured here. You find the discussion in chapter "Local fixed point theory", paragraph 7 "Parabolic fixed points: the Leau-Fatou flower". Though not for beginners its imho very insightful. jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/01/2009, 09:04 PM (08/01/2009, 08:21 AM)bo198214 Wrote: (07/31/2009, 11:09 PM)andydude Wrote: Some of us are writing an overview paper of the 5 or more methods we have discovered/developed for real analytic tetration. You might want to see where your cheta function would fit into such a paper. Yes, I absolutely second that, there is even a section about your method already in the paper, which I originally filled myself extracting your posts, however my computer crashed and I lost most of my data files. So I restarted that article again, but until now the section is empty. So if you want to participate ...The cheta + base-change approach? Or the "accelerated" method of finding the power series for Andrew's approach? (The latter is more a computational shortcut, but it gives/uses insight into the structure in the complex plane.) ~ Jay Daniel Fox andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 08/02/2009, 05:32 AM (08/01/2009, 09:04 PM)jaydfox Wrote: The cheta + base-change approach? Or the "accelerated" method of finding the power series for Andrew's approach? (The latter is more a computational shortcut, but it gives/uses insight into the structure in the complex plane.) I was referring to the cheta + base-change approach, which I do not understand. I understand your "accelerated" method since its uses the same logic as as natural/intuitive iteration, only it makes a different assumption about the "unknown coefficients". I think I could do a good job of describing this method if you want to focus on the cheta function. Andrew Robbins « Next Oldest | Next Newest »

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