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 Universal uniqueness criterion? bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/23/2008, 12:07 PM Ok, now the promised proof for the uniqueness of the gamma function, which was point 1. here. I nearly only translate it from the mentioned book: We consider the function $v=F-\Gamm$ on the right half plane, it also satisfies $v(z+1)=zv(z)$. Hence $v$ has a meromorphic continuation to $\mathbb{C}$. Poles can be at most at 0, -1, -2 , .... As $v(1)=0$ we have $\lim_{z\to 0} zv(z)=0$, hence $v$ has a holomorphic continuation to 0 and also to to each $-n$, $n\in\mathbb{N}$ by $v(z+1)=zv(z)$. $v(z)$ restricted to $1\le \Re(z)<2$ is bounded because $\Gamma$ is bounded there. Then $v(z)$ is also restricted on $S$ given by $0\le \Re(z)\le 1$. $q(z)$ defined by $v(z)v(1-z)$ is bounded on $S$ because $v(z)$ and $v(1-z)$ have the same values on $S$. Now $q(z+1)=-q(z)$, hence $q$ is bounded on whole $\mathbb{C}$ and by Liouville $q(z)=q(1)=0$. Hence $v\equiv 0$ and $F\equiv\Gamma$. « 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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