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 Iterating at fixed points of b^x bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 09/08/2007, 10:34 AM Proposition. All fixed points of $e^x$ are hyperbolic, i.e. $|\exp'(a)|\neq 0,1$ for each complex $a$ with $e^a=a$. Clearly $\exp'(a)=\exp(a)=a$. So we want to show that $|a|\neq 0,1$. We can exclude the case $|a|=0$ as this implies $a=0$ and we know that 0 is not a fixed point of $\exp(x)$. For the other case we set $a=re^{i\alpha}=r\cos(\alpha)+ir\sin(\alpha)$ and get $re^{i\alpha}=e^{r\cos(\alpha)+ir\sin(\alpha)}=e^{r\cos(\alpha)}e^{ir\sin( \alpha )}$ and hence the equation system: $\ln ( r )=r\cos(\alpha)$ (1) and $\alpha=r\sin(\alpha)$ (2). We square both equation and add them: $\ln ( r)^2+\alpha^2=r^2 \cos(\alpha)^2+ r^2\sin(\alpha)^2=r^2$ $\alpha=\pm\sqrt{r^2-\ln ( r)^2}$. But beware, this is only a necessary condition on the fixed points. The fixed points lie discretely on the complex plane. Not every point satisfying this equation is a fixed point. But from this condition we can look what happens for $|a|=r=1$. For $r=1$ we get $\alpha=\pm 1$ and we see that both values of $\alpha$ do not satisfy equation (2). So there is no fixed point with $|a|=1$. « Next Oldest | Next Newest »

 Messages In This Thread Iterating at fixed points of b^x - by bo198214 - 09/08/2007, 10:02 AM The fixed points of e^x - by bo198214 - 09/08/2007, 10:34 AM The fixed points of b^x - by bo198214 - 09/08/2007, 11:36 AM RE: Iterating at fixed points of b^x - by jaydfox - 09/12/2007, 06:23 AM RE: Iterating at fixed points of b^x - by bo198214 - 09/12/2007, 09:54 AM RE: Iterating at fixed points of b^x - by GFR - 10/03/2007, 11:03 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 06:53 AM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 01:30 PM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 05:54 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 11:05 PM RE: Iterating at fixed points of b^x - by GFR - 01/31/2008, 03:07 PM

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