07/01/2022, 08:58 PM
(07/01/2022, 03:55 AM)Daniel Wrote: Can the Riemann mapping theorem be used to convert a complex solution for tetration to a real solution? If so then the solutions for Schroeder's functional equation on my website Tetration.org can be mapped into real solutions. Since the mapping is biholomorphic then real solutions could be mapped into complex ones.
Depends what you are asking. Kneser explicitly does this procedure. If you mean for complex bases, I suggest Paulsen's paper. He describes how to analytically continue the Kneser mapping theorem from \(b > \eta\) to every base except \(0\), with a branch cut along the negative axis, (iirc).
It does use Schroder's functional equation about two fixed point pairs, \(L^+,L^-\), where for real \(b > \eta\) these are complex conjugates.