Let
$f(z)= \sum_{n=1}^{\infty} z^{(1/z)^{n}}$
A domain colored portrait (with artifacts) for $f(z)$ on the unit disk looks like:
The gray and white regions are where the software package had difficulty with the sum, because it was divergent. Is there any good way of characterizing for which $|z|<1$ this series is convergent? I'm interested in the boundary of the red dagger shaped region to the right.
The code that was used to generate this is (using mpmath and matplotlib):
from mpmath import * import pylab def f(z): return fp.nsum(lambda n: z**(1/(z**n)), [1,inf]) fp.cplot(lambda z: f(z), [-1.0, 1.0], [-1.0, 1.0], points=800000, verbose=True)