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 Universal uniqueness criterion? bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 05/22/2008, 11:25 AM andydude Wrote:So by "bounded" are you referring to the fact that Kouznetsov's extension requires that the limit towards $i\infty$ is finite? Is this the same as saying that this limit exists? His assumption is that the limit is the fixed point in the upper complex halfplane, not only finite. I mentioned somewhere already that for $1 with regular tetration this is no more true (i.e. has no limit), however it is still bounded. So perhaps this is a good generalization. Quote:Where are these proven? How are these proven? By whom? Ok, lets start with the uniqueness of $b^x$ (I dont know whether is somewhere written): Proposition.Let $b>1$ then $f(z)=b^z=e^{\ln(b)z}$ is the only holomorphic solution, defined on the right halfplane $\Re(z)\ge 0$ [this condition is not necessary, I just include it to emphasize on non-entire functions], of the equations $f(z+1)=bf(z)$, $f(0)=1$ which is bounded on the strip $S$ given by $0\le \Re(z)<1$. Proof We know that every other solution must be of the form $g(z)=f(z+p(z))$ where $p$ is a 1-periodic holomorphic function (this can roughly be seen by showing periodicity of $h(z)=f^{-1}(g(z))-z$). In this case this means: $f(z+p(z))=b^{z+p(z)}=b^{p(z)}b^z=b^{p(z)}f(z)=:q(z)f(z)$ where $q$ is also a 1-periodic function. As $g$ (and $f$) is bounded on $S$, $q$ must be bounded too. As $q$ is periodic it can be continued from $S$ to the whole plane $\mathbb{C}$ and is hence an entire holomorphic function, which is still bounded. By Liouville $q$ must be constant: $g(z)=qf(z)$. And now we apply $g(0)=1$ and see that $q=1$. I will describe the proof for the Gamma function in another post, I found it in a German complex analysis book: Reinhold Remmert, "Funktionnentheorie", Springer, 1995. As reference is given: H. Wielandt 1939. And the proof for the Fibonacci function is given in the sci.math.research thread by Waldek Hebisch (key points), and a bit fleshed out by me. « Next Oldest | Next Newest »

 Messages In This Thread Universal uniqueness criterion? - by bo198214 - 05/21/2008, 06:24 PM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 05:19 AM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 06:42 AM RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 11:25 AM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 03:11 PM RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 05:55 PM RE: Universal uniqueness criterion? - by bo198214 - 05/23/2008, 12:07 PM RE: Universal uniqueness criterion? - by Gottfried - 06/25/2008, 06:15 AM Uniqueness of analytic tetration - by Kouznetsov - 09/30/2008, 07:58 AM RE: Uniqueness of analytic tetration - by bo198214 - 09/30/2008, 08:17 AM RE: Universal uniqueness criterion? - by bo198214 - 10/04/2008, 11:19 PM RE: Universal uniqueness criterion? - by Kouznetsov - 10/05/2008, 12:22 AM RE: Universal uniqueness criterion? - by Kouznetsov - 06/19/2009, 08:45 AM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 02:04 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 02:51 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 04:19 PM RE: miner error found in paper - by bo198214 - 06/19/2009, 04:53 PM i don't think it will work - by Base-Acid Tetration - 06/19/2009, 05:17 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 06:25 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 06:27 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 07:59 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/20/2009, 02:01 PM RE: Universal uniqueness criterion? - by bo198214 - 06/20/2009, 02:10 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/23/2009, 02:39 PM RE: Universal uniqueness criterion? - by Kouznetsov - 06/23/2009, 05:46 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/23/2009, 09:28 PM RE: Universal uniqueness criterion? - by Kouznetsov - 06/24/2009, 05:02 AM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 07/04/2009, 11:17 PM RE: Universal uniqueness criterion? - by Kouznetsov - 07/05/2009, 08:28 AM RE: Universal uniqueness criterion? - by bo198214 - 07/05/2009, 06:54 PM

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