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 Dmitrii Kouznetsov's Tetration Extension bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 04/22/2008, 08:18 AM (This post was last modified: 04/25/2008, 04:14 PM by bo198214.) 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.) Corrected in the original post. Quote: How about the collaboration? That would be great. Quote:(P.S. Does the number of replies I should answer grow as the Ackermann function of time, or just exponentially?) I would guess its rather logarithmically (much questions at the start but then slowly ebbing away)! At least here is another question: 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 $b^x$. @Andrew Can you post a comparison graph with your slog/sexp? The values are in Dmitrii's paper. How does the periodicity of your slog (or was it sexp?) at the imaginary axis compare with Dmitrii's limit $\lim_{y\to\infty} F(x+iy)=L$ (where $L$ is a fixed point of $\exp$)? « 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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