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Moving between Abel's and Schroeder's Functional Equations - Printable Version

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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
Then Schroeder's equation , but is complex.  
Personally I think I prefer instead of for the complex valued Abel function.
There is a pair of complex valued Abel functions for the two complex conjugate fixed points, and there is a singularity at

Anyway, Kneser's tetration uses a Riemann mapping of , wrapping the real axis around a unit circle to eventually get to 
where there are two 1-cyclic theta(z) functions
 
where k is a constant as Im(z) gets arbitrarily large, and Kneser's slog or the inverse of Tetration would be

tau^{-1}(z) is also a z+1-cyclic function used to generate Tet(z) from the inverse of the complex valued Abel function.

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