05/29/2017, 09:14 AM
(This post was last modified: 06/01/2017, 12:59 PM by sheldonison.)
(05/28/2017, 08:46 PM)JmsNxn Wrote: Very pretty pictures.
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Define the sequence \( \zeta_n = \Psi^{-1}(e^{-n} \Psi(x_0)) \) where \( x_0 \) is fixed and is arbitrary, so long as \( \log^{\circ n}(x_0) \to L \). We can write an entire super function of exponentiation as
\( F(z)=\frac{1}{\Gamma(Lz)} (\sum_{n=0}^\infty \zeta_n \frac{(-1)^n}{n!(n+Lz)} + \int_1^\infty (\sum_{n=0}^\infty \zeta_n \frac{(-t)^n}{n!})t^{Lz-1}\,dt) \)
This expression converges FOR ALL complex values, which is REAL nice.
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There is a lot more to say about the pretty picture ... I am intrigued by the large islands of "black" which is where the \( \Psi^{-1}(z) \) function takes on nearly the value of zero, And then those nearly zero black islands lead to islands of various shades of red corresponding to values near 1, e, e^e ... until Chaos takes over and the function becomes a checkerboard of black and white with arbitrarily large values and arbitrarily small values arbitrarily close to one another. And then there is the incredibly complex singularity at \( \Psi(0) \) ....
It is interesting how \( \Gamma \) and the integral got into your equation. I should know just enough about the Gamma function to follow its derivation.
As a practical matter, I have settled on using the \( \Psi^{-1}(z) \) formal Taylor series expansion, which works well, and is quick and easy to generate in a pari-gp program. Not that math needs be practical.
For the inverse Schröder function, we are free to iterate \( z\mapsto \frac{z}{\lambda} \) as many times as we like before using the series to evaluate \( \Psi^{-1}(z)=\exp^{\circ n}\Psi^{-1}(z\lambda^{-n}) \). Then the superfunction is \( F(y)=\Psi^{-1}(\lambda^y) \)
For a 44 term expansion, you get nearly 28 decimal digits of accuracy if |z|<1.
As far as the next step, Kneser's algorithm, I think that appears to be much harder for folks to understand. So I thought I would focus on the basics of the \( \Psi \) function for awhile. I think it would help to rewrite all of the relevant equations in terms of the more accessible \( \Psi \) Schröder and inverse Schröder functions... and limit using the intermediate superfunction equation as much as possible.
- Sheldon