Fundamental Principles of Tetration

[sheldonison]

....
In what manner is Kneser's solution preferred? Please explain the mathematics behind it or provide links. My understanding is that use of Schroeder's equation and Abel's equation are exclusive. Abel's equation is for limit point \( c \) where \( f\large(c\large) = c \) and \( \left|f'\large(c\large) \right|=1. \) Schroeder's equation is for \( \left|f'\large(c\large) \right| \notin 0,1. \)

Daniel,

Henryk has a couple of posts here. First, I confess that when I wrote my first kneser.gp pari-gp program nearly six years ago, I was merely guessing at Kneser's solution, but did not understand Kneser's Riemann mapping. Later I showed that my theta mappings are equivalent to Kneser's Riemann mapping, assuming convergence.
http://math.eretrandre.org/tetrationforu...hp?tid=213
http://math.eretrandre.org/tetrationforu...hp?tid=358

Hopefully, I don't make any typos! Now, start with the two fixed points, L,L*=0.31813 +/- 1.3372i; each fixed point leads to a Schroeder equation, and its inverse. You are familiar with the Schroeder equation, lets call the Schroeder solution \( S(z) \). Call the Abel equation \( \alpha(z) \), and its inverse, \( \alpha^{-z}(z) \)

\( \alpha^{-1}(z) = S^{-1}(L^z)\;\;\; \) This is the complex valued superfunction, which you are familiar with

Now, we define two 1-cyclic \( \theta(z), \; \theta*(z) \) mappings. One theta(z) mapping is defined in the upper half of the complex plane; and the other in the lower half of the complex plane.

\( \theta(z)=\sum_{n=0}^{\infty} a_n
\cdot \exp(2n\cdot \pi i z) \)
\( \text{sexp}(z) = \alpha^{-1}_L(z+\theta(z)) \;\;\; \) we require sexp(z) be real valued at the real axis
\( \text{sexp}(z) = \alpha^{-1}_{L*}(z+\theta*(z)) \)

Now, it turns out one can write an equation for \( \theta(z) \) exactly in terms of Kneser's Riemann mapping. \( \theta(z) \) has a really nasty singularity at the real axis for integer values of z, but is analytic with no other singularities in the upper half of the complex plane, and \( \theta(z) \) decays to a constant as imag(z) goes to infinity.
\( \lim_{z \to + i\infty}\; \theta(z)=a_0=k \)

Hopefully, that's a start for you.
- Sheldon Levenstein