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 Universal uniqueness criterion? Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008   09/30/2008, 07:58 AM (This post was last modified: 09/30/2008, 07:59 AM by Kouznetsov.) Uniqueness of analytic tetration, scheme of the proof. Notations. Let $\mathbb{C}$ be set of complex numbers. Let $\mathbb{R}$ be set of real numbers. Let $\mathbb{N}$ be set of integer numbers. Let $\mathbb{C}^{\prime}=\mathbb{C} \backslash \{ z| z \in \mathbb{R}, z\le 2 \}$ Let $b>1$ be base of tetration. Assume that there exist analytic tetration $F$ on base $b$, id est, (0) $F$ is analytic at $\mathbb{C}^{\prime}$ (1) for all $z\in \mathbb{C}^{\prime}$, the relation $F(z+1)=\exp_b(F(z))$ holds (2) $F(0)=1$ (3) $F$ is real increasing function at $\{z \in \mathbb{R}| z>-2 \}$. Properties. From assumption (1) and (2) it follows, that function $F$ has singularity at -2, at -3 and so on. Consider following Assumption: There exist entire 1-periodic function $h$ such that $G(z)=F(z+h(z))$ is also analytic tetration on base $b$. Then, function $I(z)=z+h(z)$ is not allowed to take values -2, -3, .. being evaluated at elements of $\mathbb{C}^{\prime}$. This means that function $J(z)=I(z) \frac{z+2}{I(z)+2} =(z+h(z)) \frac{z+2}{z+h(z)+2}$ is entire function. (Weak statement which seems to be true) Function $J$ cannot grow faster than linear function at infinity in any direction. Therefore, it is linear function. Therefore, $h=0$ Therefore, there exist only one analytic tetration. « 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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