How do I cite this document and does it say what I think it says?

[sheldonison]

Okay, you keep mentioning over and over about this alleged method of computing such functions, but what actually are they?

I'm not sure what your question is.  Is your question how to load the program?  You would first need to download pari-gp; here is the link: https://pari.math.u-bordeaux.fr/download.html
And then my fatou.gp program is here: https://math.eretrandre.org/tetrationfor...p?tid=1017
download the program and run in it in pari-gp

Were you able to understand the descriptions in this post and the links I've posted?  Or are you asking for more specific detail?  It is important to thoroughly understand Wikipedia-Schroder's-equation as applied to the complex fixed point for "f(z)=e^z", since that is the basis upon which Kneser's Riemann mapping is built, and it is also the basis for my faou.gp program.  Then you have to understand Kneser's Riemann mapping and how it can be equivalently expressed as a 1-cyclic Fourier series that changes the complex valued Schroder function to a real valued slog.

The next paragraph is an expository summary, with no equations, to provide a brief high level overview.  Assuming you understand the background, then my fatou.gp program iterates a pair of functions, an slog Taylor series, and the 1-cyclic Fourier series theta mapping which modifies the Schroder function.   First the Schroder function is changed into the complex valued Abel function pictured in post#17.  In the picture below, my program's slog Taylor series approximation samples 60 equally spaced points around the middle circle.  The points in yellow are paired up with exp(z), one for one.  The points in brown are paired up with log(z), also one for one.  The points in pink (and green for the complex conjugate), are used to generate the 1-cyclic Fourier series theta mapping.  You can see the pink spiral towards the fixed point.  The slog has a singularity at the fixed point where it goes to imag(infinity), so the radius of convergence of the slog is the blue circle.  The pink points inside the smaller are used to generate a 1-cyclic mapping involving the Schroder equation, which is used to generate an accurate value for the pink sample points in the middle circle.  It turns out this can be either be approximately solved which is what I do, or used to generate a 60x60 simultaneous equation, which my program can also do.  These 60 sample points give more than double precision accurate results.  If you solve the simultaneous equation, then all of the points will match exactly with their yellow or brown counterparts.  The pink/green points will exactly match the 1-cyclic theta mapping.  To get more higher precision results, we keep adding more and more sample points to the circle, and to the theta mapping, to get whatever precision we would like for the slog, and for the theta mapping.

Since the Schroder equation modified by the 1-cyclic theta mapping is equivalent to Kneser's Riemann mapping, the end result is Kneser's slog calculated to whatever precision we would like. Inside the pink spiral, the Schroder equation modified by the 1-cyclic theta mapping is used to generate accurate results; inside the middle circle, the slog Taylor series gives accurate results.   Both functions give accurate results where the two representations overlap.  We can also iterate exp(z) or log(z) to get to a point represented accurately.  


There's a lot more detail.  My hope is when I have enough time, I would eventually publish a proof that the systems of equations approach using equally spaced points for both the slog taylor series, and the 1-cyclic fourier series, can be shown to be exactly Kneser's slog in the limit as the sample points get arbitrarily close together.  This requires proving the system of equations for a finite number of sample points always has exactly one solution.  And it requires showing that the error term is predictable and can be rigorously bounded.

I would also highly recommend William Paulson's work.
William Paulson has an online calculator here: http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html
William Paulson's paper's describe his algorithm: http://myweb.astate.edu/wpaulsen/tetration.html