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 tetration base conversion, and sexp/slog limit equations bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 02/21/2009, 12:39 PM (This post was last modified: 02/21/2009, 12:44 PM by bo198214.) I have only a rough idea about what you suggest, so please let me ask some more questions Let $\eta=e^{1/e}$, when we consider $\exp_\eta$ we realize that $\lim_{x\to\infty} \exp_\eta(x) = e$, and $\lim_{x\to\infty} {\exp_\eta}'(x)=0$. So the critical region is at infinity, where there $\lim_{x_0\to\infty}\exp_\eta(x_0-1) -\exp_\eta(x_0)=\log_\eta(\log_\eta(e))-\log_\eta(e)=0$. Vice versa $\lim_{x\uparrow e}\text{slog}_\eta (x)=\infty$ and $\lim_{x\uparrow e}{\text{slog}_\eta}' (x)=\infty$. I still dont see how you will use this for defining $\exp_\eta$. sheldonison Wrote:Consider what happens as b approaches e^(1/e) in the equation $\text{slog}_b(x)$. The curve becomes more and more linear, and there are fewer and fewer degrees of freedom for how to extend the sexp function to real numbers, and still have an increasing "well behaved" function. It must be possible to describe this rigorously in terms of limits. Here is an example. if $b=1.49208... \text{ slog}_b(e)=6$, $\text{slog}_b(2.4989)=5$, $\text{slog}_b(2.2887)=4$, $\text{slog}_b(2.0691)=3$, $\text{slog}_b(1.817)=2$, $\text{slog}_b(1.4923)=1$, $\text{slog}_b(1.0)=0$, What do you mean by "there are fewer and fewer degrees of freedom for how to extend the sexp function to the real numbers"? And otherwise of course you surely read already Jay's post about change of base, which attributes the same effect you describe here that $\text{slog}_a(x)-\text{slog}_b(x)$ converges/should converge to a constant for $x\to \infty$ if $a,b>\eta$. Moreover he gives a formula to derive $\text{sexp}_b$ from any given $\text{sexp}_a$ such that exactly this condition is satisfied. « 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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