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 Uniqueness summary and idea bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 08/14/2007, 09:59 AM (This post was last modified: 08/14/2007, 10:49 AM by bo198214.) Let $F(x)=b^x$ for some base $b$. Then we demand that any tetration $f(x)={}^x b$ is a solution of the Abel equation $f(x+1)=F(f(x))$ and $f(1)=b$. Such a solution $f$ (even if analytic and strictly increasing) is generally not unique because for example the solution $g(x):=f(x+\frac{1}{2\pi}\sin(2\pi x))$ is also an analytic strictly increasing solution, by $g(x+1)=f(x+1+\frac{1}{2\pi}\sin(2\pi + 2\pi x))=F(f(x+\frac{1}{2\pi}\sin(2\pi x))=F(g(x))$ and $g'(x)=f'(x+\frac{1}{2\pi}\sin(2\pi x))(1+\frac{1}{2\pi}\cos(2\pi x)2\pi)=\underbrace{f'(x+\frac{1}{2\pi}\sin(2\pi x))}_{>0}\underbrace{(1+\cos(2\pi x))}_{\ge 0}>0$ For uniqueness it was Daniel's idea to consider the continuous iteration at a fixed point $x_0$ of $F(x)$. The continuous iteration of $F$ is derived from $f$ by (0) $F^{\circ t}(x)=f(f^{-1}(x)+t)$ For such an iteration to be unique it suffices to demand the existence of the limit (1) $\lim_{x\to x0}\frac{F^{\circ t}(x)-x_0}{x-x_0}$ for the hyperbolic fixed point or (2) $\lim_{x\to x0}\frac{F^{\circ t}(x)-x}{(x-x_0)^q}$ for the parabolic fixed point (where $q>1$ is the index of the first non-zero coefficient in the development of $F$ at $x_0$). So (2) is our uniqueness condition for $b=e^{1/e}$ and (1) is our uniqueness condition for $1. Now I was thinking further that if $f_b(x)={}^xb$ is also analytic in $b$, i.e. $x\mapsto {}^t x=\exp_x^{\circ t}(1)$ is an analytic function on $(1,e^{1/e})$ (which has to be proved but is quite reasonable) then this function can be analytically extended to $x\ge e^{1/e}$ if there is no singularity at $x=e^{1/e}$. So this would mean we had a unique analytic (in $x$ and $y$) tetration ${}^y x$ under the condition (1) and (2), where $F$ is defined by (0). Note, that we dont need a converging development of $F^{\circ t}(x)$ at the fixed point $x_0$ (which for $b=e^{1/e}$, $x_0=e$, is equivalent to the existence of a converging development of $e^x-1$ at 0). The analytic function $F^{\circ t}(x)$ is uniquely determined at least for $1 and this suffices for our $F^{\circ t}(1)$. « Next Oldest | Next Newest »

 Messages In This Thread Uniqueness summary and idea - by bo198214 - 08/14/2007, 09:59 AM RE: Uniqueness summary and idea - by andydude - 08/16/2007, 05:24 AM RE: Uniqueness summary and idea - by bo198214 - 08/16/2007, 07:42 AM RE: Uniqueness summary and idea - by Gottfried - 08/16/2007, 10:37 AM RE: Uniqueness summary and idea - by bo198214 - 08/16/2007, 11:08 AM RE: Uniqueness summary and idea - by jaydfox - 08/16/2007, 06:15 PM RE: Uniqueness summary and idea - by bo198214 - 08/16/2007, 06:49 PM RE: Uniqueness summary and idea - by jaydfox - 08/16/2007, 06:58 PM RE: Uniqueness summary and idea - by bo198214 - 08/16/2007, 07:02 PM RE: Uniqueness summary and idea - by jaydfox - 08/16/2007, 09:06 PM RE: Uniqueness summary and idea - by bo198214 - 08/16/2007, 09:29 PM RE: Uniqueness summary and idea - by jaydfox - 08/16/2007, 10:03 PM RE: Uniqueness summary and idea - by bo198214 - 08/16/2007, 10:13 PM RE: Uniqueness summary and idea - by jaydfox - 08/16/2007, 10:30 PM

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