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 Fractional calculus and tetration JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 11/17/2014, 09:50 PM Well hello everybody. It's been a while since I posted here. I've been working vivaciously on iteration and fractional calculus and the ways the two intertwine and I've found a nice fact about tetration. I've been able to prove an analytic continuation of tetration for bases $1 < \alpha < e^{1/e}$ and I've been wondering about the base change formula, if this admits a solution for $\alpha > e^{1/e}$. The solution I've generated is periodic with period $2\pi i / \log(\beta)$ where $\beta$ is the attracting fixed point of $1 < \alpha < e^{1/e}$. The solution is culminated in two papers, where the first paper is purely fractional calculus and the second paper reduces problems in iteration to problems in fractional calculus. On the whole, I am able to iterate entire functions $\phi$ with fixed points $\xi_0$ such that $0<\phi'(\xi_0) < 1$. And then only able to iterate the function within the region where $\xi$ is such that $\lim_{n \to \infty}\phi^{\circ n}(\xi) = \xi_0$. For exponentiation this implies we can only iterate exponential functions with a fixed point such that its derivative is attracting and positive. This equates to functions with fixed points whose derivative is positive and less than one. From this I am wondering if it is possible to extend tetration to bases bigger than $\eta$. Since I have been able to generate tetration bases less than $\eta$ which are unique and determined by a single equation involving only naltural exponentiations of the number, $\alpha,\, \alpha^\alpha,\,\alpha^{\alpha^{\alpha}}, \,^4 \alpha,\,....$. As in we only need know the natural iterates of $\alpha$ exponentiated, to produce the complex iterates. I am mostly just wondering about the basechange formula to see if I can generate bases greater than $e^{1/e}$ using the method I've found. The benefit is that my method is unique, no other function is exponentially bounded which interpolates the values of tetration on the naturals. All I ask is if anyone can explain the base change formula clearly and if knowing tetration for bases < \eta can extend tetration for greater values. I have not written my formula for tetration as I am trying to write a paper which contains it in a simple proof. I wish to hide the simple proof ^_^. sheldonison Long Time Fellow Posts: 630 Threads: 22 Joined: Oct 2008 11/19/2014, 06:00 PM (This post was last modified: 11/19/2014, 11:42 PM by sheldonison.) (11/17/2014, 09:50 PM)JmsNxn Wrote: Well hello everybody. It's been a while since I posted here. I've been working vivaciously on iteration and fractional calculus and the ways the two intertwine and I've found a nice fact about tetration. I've been able to prove an analytic continuation of tetration for bases $1 < \alpha < e^{1/e}$ and I've been wondering about the base change formula, if this admits a solution for $\alpha > e^{1/e}$. The solution I've generated is periodic with period $2\pi i / \log(\beta)$ where $\beta$ is the attracting fixed point of $1 < \alpha < e^{1/e}$.... I just posted a similar comment on Mathstack. I like it enough to copy it here... Technically, $^x b$, for $b, The Kneser Tetration becomes ambiguous. For real bases $b>e^{1/e}$, Tetration is well defined, and analytic with singularities at negative integers<=-2. The base $b=e^{1/e}$ is the branch point, where iterating no longer grows arbitrarily large. I investigated what happens to Tetration when we extend it analytically to complex bases, and it turns out that for $be^{1/e}$, we are using both complex conjugate fixed points to generate Kneser's real valued at the real axis Tetration. And if we move the base in a circle around $e^{1/e}$ slowly using complex bases, from a real base greater than $e^{1/e}$ to one less than $e^{1/e}$, then we get to a function still uses both the attracting and repelling fixed points, but the function is no longer real valued at the real axis. Using the attracting fixed point is not the same function as Kneser's Tetration. Ok, now about the "base change" function. I could find a link on this forum, but here is a very short description: If you develop real valued Tetration by iterating the logarithm of another super-exponentially growing function, you get a function that is infinitely differentiable and looks a lot like Tetration, but it turns out to be nowhere analytic. Lets say $f(x)$ is a super-exponentially growing function, and we want to develop the "base change" Tetration for $b>e^{1/e}$ by iterating the logarithm of f(x) as follows: $\text{Tet}_b(x) \; = \lim_{n \to \infty} \log_b^{o n} \left(f(x+k_n+n)\right); \;\;\;\;\; f(k_n+n)\; = \; ^n b; \;\;\;\; k_n\;$ quickly converges to a constant as n increases If f is Tetration for another base, or many other super-exponentially growing functions, then it turns out that all of the derivatives at the real axis converge, but they eventually grow too fast for this base change Tetration function to be an analytic function. Also, the base change function is not defined in the complex plane. I haven't posted a rigorous proof of the nowhere analytic result. - Sheldon JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 11/19/2014, 10:54 PM (This post was last modified: 11/19/2014, 11:00 PM by JmsNxn.) (11/19/2014, 06:00 PM)sheldonison Wrote: (11/17/2014, 09:50 PM)JmsNxn Wrote: Well hello everybody. It's been a while since I posted here. I've been working vivaciously on iteration and fractional calculus and the ways the two intertwine and I've found a nice fact about tetration. I've been able to prove an analytic continuation of tetration for bases $1 < \alpha < e^{1/e}$ and I've been wondering about the base change formula, if this admits a solution for $\alpha > e^{1/e}$. The solution I've generated is periodic with period $2\pi i / \log(\beta)$ where $\beta$ is the attracting fixed point of $1 < \alpha < e^{1/e}$.... I just posted a similar comment on Mathstack. I like it enough to copy it here... Technically, for $^x b$, even for $b, The Kneser Tetration becomes ambiguous. For real bases $b>e^{1/e}$, Tetration is well defined, and analytic with singularities at negative integers<=-2. The base $b=e^{1/e}$ is the branch point, where iterating no longer grows arbitrarily large. I investigated what happens to Tetration when we extend it analytically to complex bases, and it turns out that for$be^{1/e}$, we are using both complex conjugate fixed points to generate Kneser's real valued at the real axis Tetration. And if we move the base in a circle around $e^{1/e}$ slowly using complex bases, from a real base greater than $e^{1/e}$ to one less than $e^{1/e}$, then we get to a function still uses both the attracting and repelling fixed points, but the function is no longer real valued at the real axis. Using the attracting fixed point is not the same function as Kneser's Tetration. Ok, now about the "base change" function. I could find a link on this forum, but here is a very short description: If you develop real valued Tetration by iterating the logarithm of another super-exponentially growing function, you get a function that is infinitely differentiable and looks a lot like Tetration, but it turns out to be nowhere analytic. Lets say $f(x)$ is a super-exponentially growing function, and we want to develop the "base change" Tetration for $b>e^{1/e}$ by iterating the logarithm of f(x) as follows: $\text{Tet}_b(x) \; = \lim_{n \to \infty} \log_b^{o n} \left(f(x+k_n+n)\right); \;\;\;\;\; f(k_n+n)\; = \; ^n b; \;\;\;\; k_n\;$ quickly converges to a constant as n increases If f is Tetration for another base, or many other super-exponentially growing functions, then it turns out that all of the derivatives at the real axis converge, but they eventually grow too fast for this base change Tetration function to be an analytic function. Also, the base change function is not defined in the complex plane. I haven't posted a rigorous proof of the nowhere analytic result. AHA! Thank you for the clarification! So I have proven a unique periodic extension of tetration for bases between 1 and eta with an attracting fixed point. My extension $F$ is also the sole extension that is bounded by $|F(z)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|}$ where $\rho, \alpha, C \in \mathbb{R}^+$ and $\alpha < \pi/2$. What you said concurs with what I suspected however, that since my fractional calculus iteration only works on attracting fixed points it means its useless for bases greater than eta. Crud! Nonetheless I'll post my extension soon. I just need to iron out all the wrinkles in my article. I'm trying to tie this proof with all my other fractional calculus proofs (continuum sums, the differsum, and assorted recursion concepts). fivexthethird Junior Fellow Posts: 9 Threads: 3 Joined: Nov 2013 11/20/2014, 02:56 AM (11/19/2014, 10:54 PM)JmsNxn Wrote: My extension $F$ is also the sole extension that is bounded by $|F(z)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|}$ where $\rho, \alpha, C \in \mathbb{R}^+$ and $\alpha < \pi/2$. The regular iteration for bases $1 satisfies that, as it is periodic and bounded in the right halfplane. What complex bases does it work for? Does it work for base eta? sheldonison Long Time Fellow Posts: 630 Threads: 22 Joined: Oct 2008 11/20/2014, 04:42 AM (This post was last modified: 11/20/2014, 04:46 AM by sheldonison.) (11/20/2014, 02:56 AM)fivexthethird Wrote: (11/19/2014, 10:54 PM)JmsNxn Wrote: My extension $F$ is also the sole extension that is bounded by $|F(z)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|}$ where $\rho, \alpha, C \in \mathbb{R}^+$ and $\alpha < \pi/2$. The regular iteration for bases $1 satisfies that, as it is periodic and bounded in the right halfplane. What complex bases does it work for? Does it work for base eta?The Schroeder equations that give the formal solution for attracting (and repelling) fixed points do not work for the parabolic case, base eta=e^(1/e). See Will Jagy's post on mathstack. I have written pari-gp program that implements Jean Ecalle's formal Abel Series, Fatou Coordinate solution for parabolic points with multiplier=1; this is an asymptotic non-converging series, with an optimal number of terms to use. To get more accurate results, you may iterate f or $f^{ -1}$ a few times before using the Abel series. - Sheldon JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 11/20/2014, 11:16 PM (11/20/2014, 02:56 AM)fivexthethird Wrote: (11/19/2014, 10:54 PM)JmsNxn Wrote: My extension $F$ is also the sole extension that is bounded by $|F(z)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|}$ where $\rho, \alpha, C \in \mathbb{R}^+$ and $\alpha < \pi/2$. The regular iteration for bases $1 satisfies that, as it is periodic and bounded in the right halfplane. What complex bases does it work for? Does it work for base eta? Quite literally only for those real bases so far. I'm thinking there might be a way to retrieve it for other bases, but that would require a lot of generalizing on the bare machinery I have now--making the fractional calculus techniques apply on repelling fixed points. All in all, the method I have only works for $1, and since the usual iteration is periodic and bounded like that, it must be mine as well. Currently I'm looking at how more regularly behaved functions look when they're iterated using FC--maybe that'll help me draw some more conclusions. « Next Oldest | Next Newest »

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