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

[bo198214]

there is an article of L Euler, which seems to deal with it.

Wow, Gottfried! Where did you dig out this article?!
However it seems to be beneficial being able to read Latin ...
At least I see \( e^{\frac{\pi}{2}i}=i \) in the text ...

And there we have it already, Ivar.
One branch of the LambertW function gives \( W(-\frac{\pi}{2})=-\frac{\pi}{2}i \), hence for this branch \( h(e^{\frac{\pi}{2}})=i \). (Forget what I said about \( h(e^{-\frac{\pi}{2}}) \), that was a somewhat misled speculation).

We can imagine the branches of \( h(b) \) given by the fixed points of \( b^z \).