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 tetration base conversion, and sexp/slog limit equations bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/27/2009, 03:39 PM (This post was last modified: 02/27/2009, 03:47 PM by bo198214.) I was thinking over the topic and try to put it on its feet. The paradigm is that $\lim_{x\to\infty}\text{slog}_a(\text{sexp_b}(x)) - x$ shall exist for each $a,b>\eta$. So if this limit exists, we let $x=x+n$, $a=e$ $c_{a,b}=\lim_{n\to\infty}\text{slog}_a(\text{sexp}_b(x+n)) - (x+n)$ $\lim_{n\to\infty}\log_b^{\circ n}(\text{sexp}_a(x+n+c_{a,b}))=\text{sexp}_b(x)$ (Jay's change of base formula) $\lim_{n\to\infty}\log_b^{\circ n}(\text{sexp}_a(x+n))=\text{sexp}_b(x-c_{a,b})$ $\lim_{n\to\infty}\log_b^{\circ n}({\exp_a}^{\circ n}(y)) =\text{sexp}_b(\text{slog}(y)-c_b)$ $\kappa_{b,a}(y) = \lim_{n\to\infty} {\log_b}^{\circ n}({\exp_a}^{\circ n}(y))$, $\kappa_{a,b}={\kappa_{b,a}}^{-1}$ Change of base is just the application of a function (I think one could show that $\kappa$ is analytic): $\text{slog}_b(\kappa_{b,a}(x)) = \text{slog}_a(x)-c_{a,b}$ $\kappa_{b,a}(\text{sexp}_a(x))=\text{sexp}_b(x-c_{a,b})$ While $\kappa_{a,b}$ does not depend on any $\text{sexp}$ or $\text{slog}$, $c_{a,b}$ does. we can define it by setting $x=c_{a,b}$ in the second equation: $\kappa_{b,a}(\text{sexp}_a(c_{a,b}))=1$ $c_{a,b}=\text{slog}_a(\kappa_{a,b}(1))$. So we need to show in your case that the limit: $\lim_{b\to\eta^+}\text{slog}_b(\kappa_{b,a}(x))$ exists for $x$ in some initial range, where $\text{slog}_b$ is your linear approximation. To show this we could use the Cauchy criterion. It should work in the form $\lim_{b\to\eta} f_b$ exists, if for each $\epsilon>0$ there exists a $\delta>0$ and a $b_0>\eta$such that for all $\eta < b,b' and $|b-b'|<\delta$: $|f_b\circ f_{b'}^{-1}-\text{id}|<\eps$. Where we put $f_b=\text{slog}_b\circ \kappa_{b,a}$. Suprisingly but happily the compositional difference is independent on $a$: $f_b\circ f_{b'}^{-1} = \text{slog}_b\circ \kappa_{b,a}\circ \kappa_{a,b'}\circ \text{sexp}_b = \text{slog}_b\circ \kappa_{b,b'}\circ \text{sexp}_b$. Well, I dont pretend that it helps you, but it helped me at least somewhat in understanding « Next Oldest | Next Newest »

 Messages In This Thread tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/18/2009, 07:01 AM RE: tetration base conversion, questions and results - by sheldonison - 02/19/2009, 12:10 AM tetration base conversion, uniqueness criterion? - by bo198214 - 02/19/2009, 04:24 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/20/2009, 10:54 AM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/20/2009, 01:07 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/20/2009, 02:51 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/21/2009, 12:18 AM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/21/2009, 12:39 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/21/2009, 02:59 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/21/2009, 06:36 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/22/2009, 04:41 AM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/22/2009, 04:04 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/24/2009, 08:24 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 09:57 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 10:21 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/24/2009, 10:54 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 11:06 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/26/2009, 11:04 AM RE: tetration base conversion, and sexp/slog limit equations - by bo198214 - 02/26/2009, 12:16 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/26/2009, 02:36 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/28/2009, 05:56 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/28/2009, 10:01 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/01/2009, 12:18 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 06:15 PM RE: tetration base conversion, and sexp/slog limit equations - by bo198214 - 03/03/2009, 06:46 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 07:27 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/09/2009, 06:34 PM Summay tetration base conversion, and sexp/slog limit equations - by sheldonison - 07/31/2009, 06:55 PM RE: Summay tetration base conversion, and sexp/slog limit equations - by sheldonison - 08/01/2009, 10:32 AM Is it analytic? - by sheldonison - 12/22/2009, 11:39 PM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/25/2009, 08:51 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 12/26/2009, 01:44 AM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/26/2009, 01:54 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 12/27/2009, 06:53 AM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/31/2009, 11:45 PM Inherent ringing in tetration, re: base conversion - by sheldonison - 01/02/2010, 05:31 AM RE: Inherent ringing in tetration, base conversion - by mike3 - 01/04/2010, 03:51 AM RE: Inherent ringing in tetration, base conversion - by sheldonison - 01/04/2010, 06:08 AM RE: tetration base conversion, and sexp/slog limit equations - by tommy1729 - 02/26/2013, 10:47 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/27/2013, 07:05 PM

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