# Tetration Forum

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(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