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 Universal uniqueness criterion? bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 10/04/2008, 11:19 PM (This post was last modified: 10/05/2008, 12:43 AM by bo198214.) Ok, I give it another try: Proposition. Let $S=\{z:0<\Re(z)\le 1\}$, $S_\epsilon=\{z:0<\Re(z)<1+\epsilon\}$, $\epsilon>0$ and $D\supseteq S_\epsilon$ being a domain (open and connected) of definition and $H\supseteq S_\epsilon$ being a domain (open and connected) of values for a holomorphic function. Let $f$ be a holomorphic function on $D$, $G:=f(S_\epsilon)\subseteq H$, such that (0) $H\subseteq f(D)$. (1) $f(0)=1$ (2) $f(z+1)=F(f(z))$ (U) $f^{-1}(G)$ has bounded real part. Then $g=f$ for every other on $D$ holomorphic $g$ satisfying (0), (1), (2), (U). Proof. $h(z):=g^{-1}(z)-f^{-1}(z)$ has bounded real part on $G$. We consider $f^{-1}$ and $g^{-1}$ and so $h$ to be holomorphic on the same Riemann surface $G$. $\delta(z):=h(f(z))=g^{-1}(f(z))-z$ is a 1-periodic function, holomorphic on $S_\epsilon$. As $S_\epsilon\supset S$ it can be continued to an entire function, so it has to take on every complex value with at most one exception already on the strip $S$ otherwise it is a constant. Now $\delta(S)=\delta(S_\epsilon)=h(G)$ has bounded real part and hence can not take on every value, so $h(z)=0$ and $g=f$. « 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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