Taylor series of cheta

[sheldonison]

Hello, Sergo!
Well, this programme looks really promising. Thank you for sharing it with me. It can calculate hyperops with fractional and complex ranks ... BUT it cannot calculate tetrations like H(2;4;3) which would be 16, but it gives 11.713... Why?
And I am really interested in that how can this programme evaluate Cheta and Inverse-cheta functions for fractional and complex arguments. Can you tell me it?

Here's another interesting code fragment which implements
\( f(z)=a^z;\;\;\; h(a,b,c)=f^{\circ b} ( c ) \)
This code fragment works with the other tetration program I wrote; fatou.gp The function h(a,b,c) is an analytic function in all three variables; (a,b,c); there is a parabolic singularity at \( a=\eta=\exp(1/e) \); analytic continuation explains why the sexp function is no longer real valued for fractional iterates for bases<eta.
\( h(a,b,1) = \text{sexp_a}(b) \)
\( h(a,1,b) = a^b \)
\( h(a,-1,b) = \log_a(b) \)
The half iterates of exp and log would be:
\( h(a,0.5,h(a,0.5,b)) = a^b \)
\( h(a,-0.5,h(a,-0.5,b)) = \log_a(b) \)

Code:

\r fatou.gp
h(a,b,c) = {
quietmode=1;
sexpinit(a);
return(sexp(slog(c)+b));
}