Kneser method question

[sheldonison]

Thanks. It work when I reinstall the office.

I have some questions.
(1)Can we use \( \Psi \)Schröder and inverse Schröder functions, rewrite the Kneser’s superroot?
(2)Why fatou.gp can't "recursion" well-behaved region to fix the ill-region? sexp can work well when small Imag(z), and pent/hex can work well when small Real(z).

If you have time plaese fix the bug report. It feels so bad for pent(3) can't evaluate, It just a simple imput

I'm focused on rigorously mathematically proving a simplified version of fatou.gp converges to Kneser's slog for real base e, so I'm not working on fixing any other boundary conditions for fatou.gp

You can get an Abel function from the Psi function, by taking the log_lambda
of the Psi function, where \( \lambda \) is the derivative at the fixed point; this
works for any function for which you have Psi.

\( \alpha(z)=\frac{\ln(\Psi(z))}{\ln(\lambda)} \)
But then to get Kneser's slog, you would need Kneser's Tau function; where theta (z) is a 1-cyclic function;  \( \tau(z)=z+\theta(z) \); otherwise as you are well aware, it would be a completely different function from Kneser's slog, and would give different results.  For example, \( \alpha \) would not be real valued at the real axis for real bases>exp(1/e).  

\( \text{slog}(z)=\tau(\alpha(z)) = \alpha(z)+\theta(\alpha(z)) \)
Generating \( \Psi(z) \) is fairly straight forward, but generating \( \tau(z) \) is equivalent to generate Kneser's Riemann mapping.