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 the inconsistency depending on fixpoint-selection bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 03/04/2008, 11:17 AM As I just read in Knoebel's Exponential Reiterated there is even a parametrization of the curve $f$, already given by Goldbach: Knoebel considers the equation $x^y=y^x$ which is equivalent to $x^{1/x}=y^{1/y}$ which means that the fixed points $x$ and $y$ have the same base. So $y=f(x)$. The parametrization is: $x=s^{1/(s-1)}$ and $y=s^{s/(s-1)}$. We can easily verify that this indeed satisfy $x^y=y^x$: $x^y=s^{s^{s/(s-1)}/(s-1)}=s^{s^{1+\frac{1}{s-1}}/(s-1)}=s^{s x/(s-1)}=y^x$ For example for $s=2$ we get our famous fixed points $x=2^{1/\left(2-1\right)}=2$ and $y=2^{2/\left(2-1\right)}=2^{2}=4$ There is lots of other interesting stuff in Knoebel's article, but read it yourself For our consideration here let: $f_1(s)=s^{1/(s-1)}$ and $f_2(s)=s^{s/(s-1)}$ so that $f=f_2\circ f_1^{-1}$. So I wonder whether we can express $f_1^{-1}$ ($f_1 : (0,\infty)\to(1,\infty)$ and $f_2 : (0,\infty)\to(1,\infty)$ are indeed bijective) with the Lambert W function. Any ideas? « Next Oldest | Next Newest »

 Messages In This Thread the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 03:06 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 05:20 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 10:14 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:17 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:25 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:32 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:43 AM RE: the inconsistency depending on fixpoint-selection - by GFR - 02/07/2008, 09:02 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 09:49 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/03/2008, 11:20 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 04:06 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/06/2008, 04:58 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 05:25 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 03:16 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 10:56 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 07:17 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 08:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 09:38 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:22 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:52 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 04:21 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/09/2008, 09:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 02:10 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 04:41 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 07:19 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 04/29/2008, 08:15 AM

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