Tetration Forum

Full Version: Moving between Abel's and Schroeder's Functional Equations
You're currently viewing a stripped down version of our content. View the full version with proper formatting.
(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/tetrationfor...hp?tid=213
https://math.stackexchange.com/questions...55#2308955