01/07/2020, 03:55 PM
Daniel
Moving between Abel's and Schroeder's Functional Equations

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 1cyclic 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+1cyclic 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

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