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 Andrew Robbins' Tetration Extension tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 08/23/2009, 02:45 PM (08/07/2007, 04:38 PM)bo198214 Wrote: I just read Andrew Robbins' solution to the tetration problem, which I find very convincing, and want to use the opportunity to present and discuss it here. The solution ${}^y x$ satisfies the 2 natural conditions 1. ${}^1b=b$ and ${}^{x+1}b=b^{{}^xb}$ 2. $x\mapsto b^x$ is infinitely differentiable. For $k\ge 1$ the constant -1 vanishes and we make the following calculations: $s^{(k)}(x)=(t(b^x))^{(k)}=\left(\sum_{i=0}^\infty \nu_i \frac{b^{xi}}{i!}\right)^{(k)}=\sum_{i=0}^\infty\frac{\nu_i}{i!}(b^{xi})^{(k)}$ The derivation of $b^{xi}$ is easily determined to be $(b^{xi})'=b^{xi}\text{ln}(b) i$ and so the k-th derivative is $(b^{xi})^{(k)} = b^{xi}(\text{ln}(b)i)^k$, which give us in turn $\nu_k=s(x)^{(k)}= \text{ln}(b)^k\sum_{i=0}^\infty\nu_i\frac{i^k}{i!}$ for $k\ge 1$. notice in the last line bo wrote s(x) instead of s(0). it is an expansion at x = 0. now if we consider expansions at both x = 0 and x = 1 and get the same coefficients for x = 0 by computing them from 1) the coefficients expanded at x = 1 2) solving the modified equation ( see below) then that probably means we have radius 1 or larger ( radius from the origin at x = 0 ) since b^1 i = b^i , we get an extra b^i factor on the right side. and v_k is replaced by sum v_k / k! that equation should be solvable and have the same solutions v_k IF Andrew's slog has a radius 1 ( or larger ) from the origin. bo mentioned the potential non-uniqueness for v_k when expanded at x = 0. maybe this could be the extra condition we(?) are looking for. Regards tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread Andrew Robbins' Tetration Extension - by bo198214 - 08/07/2007, 04:38 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/18/2007, 08:20 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/19/2007, 09:50 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/20/2007, 02:22 PM RE: Andrew Robbins' Tetration Extension - by andydude - 11/12/2007, 08:43 AM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 06/26/2009, 10:51 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 06/27/2009, 09:39 AM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 06/28/2009, 12:08 AM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/06/2007, 04:17 AM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/06/2007, 04:27 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/06/2007, 10:57 AM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/06/2007, 01:58 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/06/2007, 03:58 PM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/12/2007, 09:14 AM RE: Andrew Robbins' Tetration Extension - by andydude - 11/12/2007, 09:56 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/12/2007, 08:05 PM RE: Andrew Robbins' Tetration Extension - by andydude - 11/13/2007, 12:16 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/13/2007, 10:21 AM RE: Andrew Robbins' Tetration Extension - by andydude - 11/13/2007, 05:45 PM RE: Andrew Robbins' Tetration Extension - by Gottfried - 03/17/2008, 07:52 AM RE: Andrew Robbins' Tetration Extension - by Gottfried - 03/17/2008, 06:09 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 06/29/2009, 08:20 PM RE: Andrew Robbins' Tetration Extension - by andydude - 07/27/2009, 08:10 AM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/11/2009, 12:18 PM RE: Andrew Robbins' Tetration Extension - by jaydfox - 08/11/2009, 07:06 PM RE: Andrew Robbins' Tetration Extension - by jaydfox - 08/11/2009, 07:12 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/23/2009, 02:45 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/23/2009, 03:23 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/26/2009, 04:01 PM RE: Andrew Robbins' Tetration Extension - by andydude - 09/04/2009, 06:42 AM RE: Andrew Robbins' Tetration Extension - by Gottfried - 12/28/2009, 05:21 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/18/2016, 12:29 PM RE: Andrew Robbins' Tetration Extension - by Gottfried - 08/22/2016, 04:19 PM

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