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 Universal uniqueness criterion? Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 06/22/2009, 12:37 AM (This post was last modified: 06/22/2009, 04:49 AM by Base-Acid Tetration.) Please go back up to the beginning of the proof; I made the conditions a lot stronger: changed the holomorphy condition for $f$ to BIholomorphy on a strip with infinite range of real parts), and added "f has no other fixed points", because I thought other fixed points would mess things up. Proof continued... 4. Consider a simple curve $\gamma \subset D_1$ that also has $L$ and $\bar{L}$ as non-inclusive boundaries. (1) Since $A_1$ is biholomorphic on $\gamma$, $A_1$ will still be biholomorphic on the curve $f(\gamma)$, because if $A_1(z)$ is biholomorphic, so is $A_1(z)+1=A_1(f(z)) \forall z \in D_1$. (2) We can do this for every curve in $D_1$ that has $L$ and $\bar{L}$ as boundaries, to extend the domain of $A_1$ (3) We can repeat (4.2) as many times as needed to get to $D_2$. (we can do the above the other way around, using $f^{-1}$ to get from $D_2$ to $D_1$, because f is BIholomorphic) Then $A_1$ is biholomorphic where $A_2$ was defined to be biholomorphic.* Then we know, by the theorem that was proven on the other day, that they are the same biholomorphism. So we know that there exists a single open set C that includes $D_1 \cup D_2$ where not only the biholomorphism $A_{cont}$that maps d to c exists, but also $A_1 = A_2 = A_{cont} \forall z \in C.$ $\mathcal{Q. E. D.}$ Corollary. There exists a unique superlogarithm for $b>e^{1/e}$ that uniquely bijects holomorphically each simple initial region of arbitrary "width" in an open set C that: (1) contains in its boundary fixed points of exp_b; (2) does not include branch cuts of the superlogarithm; to its resp. vertically infinite strip of arbitrary width in $A( C )$. *Now I don't know if $A_1 = A_2$ in $D_2$. It must have something to do with the condition $A(d) = c$. For your theorem to apply, I need to prove that both A1 and A2 are equal in some neighborhood of d. « 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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