Fundamental Principles of Tetration

[sheldonison]

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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

... Now, it turns out one can write an equation for \( \theta(z) \) exactly in terms of Kneser's Riemann mapping.

Actually, I find it easiest to write an equation for Kneser's Riemann mapping in terms of my \( \theta(z) \) mapping; Here are the relevant equations:

If you start with the real valued solution, sexp(z), then you can generate Kneser's Riemann mapping as follows in terms of the complex valued Abel function, \( \alpha(z) \)
\( f(z) = \alpha(\text{sexp}(z)) = z +\theta(z)\;\; \) here, \( \theta(z) \) is my theta(z) mapping with trivial algebra since
\( \text{sexp}(z) = \alpha^{-1}(z+\theta(z)) \;\;\; \) this is my theta mapping for sexp(z)

Then Kneser's Riemann mapping results in \( \exp(2\pi i \cdot f(z)) \) where z has been wrapped around a unit circle by using the substitution \( z = \frac{\log(y)}{2\pi i} \).

Then Kneser's Riemann mapping results in this unit circle function in terms of \( f(z) = z +\theta(z) \)

\( \exp \left[ 2\pi i \cdot f\left(\frac{\log(y)}{2\pi i} \right) \right]
\;\;\; \) This is Kneser's Riemann mapping on a unit circle in terms of y and \( f(z) = z +\theta(z) \)

From there, Kneser works backwards to \( f(z) = z+\theta(z)\;\;\;\text{sexp}(z)=\alpha^{-1}(f(z))\;\;\; \) Kneser's real valued sexp(z) function using my notation