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+--- 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