Moving between Abel's and Schroeder's Functional Equations - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Moving between Abel's and Schroeder's Functional Equations (/showthread.php?tid=1251) Moving between Abel's and Schroeder's Functional Equations - Daniel - 01/07/2020 Check out Moving between Abel's and Schroeder's Functional Equations RE: Moving between Abel's and Schroeder's Functional Equations - sheldonison - 01/16/2020 (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/tetrationforum/showthread.php?tid=213 https://math.stackexchange.com/questions/2308409/operational-details-implementation-of-knesers-method-of-fractional-iteration/2308955#2308955