Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e))

[bo198214]

so h(e^-pi/2) = -i-2pik
or h(e^-pi/2 = +i+2pik

...

Or do I make a mistake somewhere?

Yes

Different branches are not necessarily \( 2\pi i k \) apart. Though this is true for the logarithm, it is for example not true for \( h \) or \( W \).
To compute (a closed form of) the power series coefficients of \( W \) at say -1 depending on the branch, seems a rather laborious task.
I am not sure how Mathematica and Maple compute the Lambert W function outside the convergence radius of its standard power series.
Perhaps they dont use a power series expansion but an iteration formula.
For example the different branches of \( h \) can be computed by infinitely applying the logarithm: \( h(b)=\lim_{n\to\infty}\log_b^{\circ n} (x_0) \) (corresponding with the branches of the logarithm) or as we know already by infinitely applying the exponential \( h(b)=\lim_{n\to\infty}\exp_b^{\circ n} (0) \) for the range \( e^{-e}<b<e^{1/e} \).

In this post I showed the dependency of the fixed points of \( b^z \) from \( b \). The fixed points of \( b^z \) are the different branches of \( h(b) \).