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 Moving between Abel's and Schroeder's Functional Equations Daniel Fellow Posts: 91 Threads: 33 Joined: Aug 2007 01/07/2020, 03:55 PM sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 01/16/2020, 10:08 PM (This post was last modified: 01/17/2020, 04:09 PM by sheldonison.) (01/07/2020, 03:55 PM)Daniel Wrote: Check out Moving between Abel's and Schroeder's Functional Equations Hey Daniel, what if $b>\exp(\frac{1}{e})$ Then Schroeder's equation $\Psi\circ b^z=\lambda\Psi$, but $\lambda$ is complex.   Personally I think I prefer $\alpha(z)$ instead of $\Phi(z)$ for the complex valued Abel function. $\alpha=\frac{\ln\Psi}{\ln \lambda};\;\;\alpha;\;\alpha^*$ There is a pair of complex valued Abel functions for the two complex conjugate fixed points, and there is a singularity at $\alpha(0,1,e,...)$ Anyway, Kneser's tetration uses a Riemann mapping of $\exp(\2\pi i(\alpha\circ\Re))$, wrapping the real axis around a unit circle to eventually get to  $\tau(z)=z+\theta_s(z);\;\;\;\tau^{-1}(z)=z+\theta_t(z)$ where there are two 1-cyclic theta(z) functions $\lim_{\Im(z)\to\infty}\theta(z)=k;\;$  where k is a constant as Im(z) gets arbitrarily large, and Kneser's slog or the inverse of Tetration would be $\text{slog}_k(z)=\tau(\alpha(z))=\alpha(z)+\theta_s(\alpha(z))$ tau^{-1}(z) is also a z+1-cyclic function used to generate Tet(z) from the inverse of the complex valued Abel function. $\text{Tet}_k(z)=\alpha^{-1}(\tau^{-1}(z))=\alpha^{-1}(z+\theta_t(z))$ https://math.eretrandre.org/tetrationfor...hp?tid=213 https://math.stackexchange.com/questions...55#2308955 - Sheldon « Next Oldest | Next Newest »

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