Base -1

[sheldonison]

Here is what analytic sexp base(-1) looks like at the real axis.

Wonderful!

Did you tried negative bases close to zero?

Yes, though I'm not really getting useful results yet.

The algorithm works with \( k=\ln(\ln(b))+1 \)
For b=0.5, that would be \( k\approx 0.633487+\pi i \), where we iterate \( z \mapsto \exp(z)-1+k \). Right now, the closest I can get to tetration base=0.5 with good convergence is b=0.5+0.39i, which is k=0.782+2.173i. I'm having several different kinds of problems with these bases, though I hope I will eventually get full convergence for some bases with imag(k)=pi, though convergence will still be limited for bases with indifferent fixed points like \( b=\exp(-e) \)