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 Dmitrii Kouznetsov's Tetration Extension Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 04/24/2008, 01:02 AM (This post was last modified: 04/24/2008, 01:04 AM by Kouznetsov.) bo198214 Wrote:andydude Wrote:See Analytic solution of F(z+1)=exp(F(z)) in complex z-plane for more information.About the uniqueness: It is well known that if we have a solution $\alpha$ of the Abel equation $\alpha(z+1)=f(\alpha(z))$ then for any 1-periodic function $\phi$ also $\beta(z)=\alpha(z+\phi(z))$ is a solution to the Abel equation. (Because $\beta(z+1)=\alpha(z+1+\phi(z+1))=\alpha(z+\phi(z)+1)=f(\alpha(z+\phi(z))=f( \beta( z))$). So let $F$ be one solution of (*) $F(z+1)=\exp(F(z))$ with (**) $\lim_{y\to\infty} F(x+iy) = L$ and $\lim_{y\to -\infty} F(x+iy)=L^\ast$ then $G(z)=G(z+\sin(z))$ is another solution of (*). Let us now consider (**). We know that $\sin(z)=-i\frac{e^{iz}-e^{-iz}}{2}$ and $\sin(x+iy)=i\frac{-e^{ix}e^{-y}+e^{-ix}e^y}{2}$ $G(x+iy)=F\left(x+iy+i\frac{-e^{ix}e^{-y}+e^{-ix}e^y}{2}\right)$. As $e^{-y}\to 0$ at least for x=0 also $\lim_{y\to\infty} G(iy)=\lim_{y\to\infty} F(i(y+e^y))=L$ bo198214 Wrote:Kouznetsov Wrote:By the way, your deduction gives the hint, how to prove the Theorem 1. Perhaps. I am not convinced yet that it is true, but if it was, this would be great. We even know vice versa if we have two solutions $F$ and $G$ of the Abel equation then $G^{-1}(F(x+1))-(x+1)=G^{-1}(\exp(F(x)))-(x+1)=G^{-1}(\exp(G(G^{-1}(F(x)))-(x+1)=G^{-1}(F(x))+1-(x+1)=G^{-1}(F(x))-x$, meaning that $\phi=G^{-1}\circ F-\text{id}$ is a 1-periodic function, so we know already that each other solution $G$ of the Abel equation must be of the form $F(z+\phi(z))$ for some 1-periodic $\phi$. To prove theorem 1 everything depends on the behaviour of those 1-periodic functions for $\Im(z)\to\infty$. For uniqueness roughly the real value must be go to infinity for the imaginary argument going to infinity. Quote:(However, we have to scale the argument of sin function.) Yes, my negligence. The $z$ in $\sin(z)$ has to be replaced by $\frac{z}{2\pi}$ in the previous post. Quote: How about the collaboration? That would be great. Can you please compute values on the real axis for bases $b for example for $b=\sqrt{2}$? I would like to compare your solution with the regular tetration developed at the lower real fixed point (which would be 2 in the case of $b=\sqrt{2}$) of $b^x$. Bo, I got your message about base b=e^(1/e) and b=sqrt(2). In these cases, the real part of quasiperiod is zero, and I cannot run my algorithm as is. I need to adopt it. It will take time. I do not think that b=sqrt(2) is of specific interest (just integer L(b)=4); we need to consider the general case. You may advance faster than I do. You may begin with the plot of the asymptotic period T(b) and analysis of its limiting behavior in vicinity of b=1 and b=e^(1/e). Please, provide the good approximation for (at least) the leading terms. P.S. you may also correct misprints in your post: invert the scaling factor for the argument of sin, and delete the expression with unmatched parenthesis. « Next Oldest | Next Newest »

 Messages In This Thread Dmitrii Kouznetsov's Tetration Extension - by andydude - 04/16/2008, 10:16 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/21/2008, 10:41 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 04/22/2008, 12:59 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/22/2008, 08:18 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 04/24/2008, 01:02 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/25/2008, 04:06 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 04/26/2008, 02:12 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/26/2008, 06:26 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/17/2008, 03:22 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/18/2008, 05:31 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/18/2008, 05:09 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/21/2008, 12:20 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/21/2008, 06:22 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/21/2008, 11:18 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/22/2008, 07:12 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/22/2008, 10:43 PM RE: Dmitrii Kouznetsov's Tetration Extension - by andydude - 05/22/2008, 10:59 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/22/2008, 11:36 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 06:21 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 08:48 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 10:09 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 02:15 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 03:47 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 04:35 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 05:52 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 11:03 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/24/2008, 05:36 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/24/2008, 09:43 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/24/2008, 09:53 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/24/2008, 11:24 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/24/2008, 11:39 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/24/2008, 12:08 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/26/2008, 07:01 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/26/2008, 09:03 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/27/2008, 03:58 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/28/2008, 08:58 AM compare complex plot with matrix power method - by bo198214 - 10/09/2008, 10:21 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 10/10/2008, 01:17 AM [split] Taylor series of upx function - by Kouznetsov - 11/20/2008, 01:31 AM

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