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 A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov Junior Fellow Posts: 12 Threads: 3 Joined: Dec 2011 10/30/2016, 11:02 PM (This post was last modified: 10/30/2016, 11:08 PM by Vladimir Reshetnikov.) After some study of different approaches to an extension of tetration to fractional or complex heights, and many numeric experiments, I came to the following conjecture, that I am currently trying to prove: Let $a$ be a fixed real number in the interval $1 < a < e^{1/e}$. There is a unique function $f(z)$ of a complex variable $z$, defined on the complex half-plane $\Re(z) > -2$, and satisfying all of the following conditions: * $f(0) = 1$. * The identity $f(z+1) = a^{f(z)}$ holds for all complex $z$ in its domain (together with the first condition, it implies that $f(n) = {^n a}$ for all $n \in \mathbb N$). * For real $x > -2, \, f(x)$ is a continuous real-valued function, and its derivative $f'(x)$ is a completely monotone function (this condition alone implies that the function $f(x)$ is real-analytic for $x > -2$). * The function $f(z)$ is holomorphic on its domain. Please kindly let me know if this conjecture has been already proved, or if you know any counter-examples to it, or if you have any ideas about how to approach to proving it. « Next Oldest | Next Newest »

 Messages In This Thread A conjectured uniqueness criteria for analytic tetration - by Vladimir Reshetnikov - 10/30/2016, 11:02 PM RE: A conjectured uniqueness criteria for analytic tetration - by sheldonison - 11/01/2016, 02:08 AM RE: A conjectured uniqueness criteria for analytic tetration - by tommy1729 - 11/30/2016, 01:26 AM RE: A conjectured uniqueness criteria for analytic tetration - by sheldonison - 12/01/2016, 12:59 PM RE: A conjectured uniqueness criteria for analytic tetration - by tommy1729 - 12/07/2016, 01:25 PM RE: A conjectured uniqueness criteria for analytic tetration - by Vladimir Reshetnikov - 01/11/2017, 12:03 AM RE: A conjectured uniqueness criteria for analytic tetration - by Vladimir Reshetnikov - 01/15/2017, 06:19 PM RE: A conjectured uniqueness criteria for analytic tetration - by JmsNxn - 01/23/2017, 07:03 AM RE: A conjectured uniqueness criteria for analytic tetration - by JmsNxn - 01/24/2017, 08:30 PM RE: A conjectured uniqueness criteria for analytic tetration - by Vladimir Reshetnikov - 01/25/2017, 12:44 AM RE: A conjectured uniqueness criteria for analytic tetration - by bo198214 - 01/25/2017, 06:26 PM RE: A conjectured uniqueness criteria for analytic tetration - by tommy1729 - 02/03/2017, 01:23 PM RE: A conjectured uniqueness criteria for analytic tetration - by JmsNxn - 01/25/2017, 09:46 PM RE: A conjectured uniqueness criteria for analytic tetration - by JmsNxn - 02/17/2017, 05:21 AM

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