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

[bo198214]

Though it is not mentioned on Mathworld one can probably use the quite easy iteration formula:
\( W(y)=\lim_{n\to\infty} f_y^{\circ n}(x_0) \) with \( f_y(x)=\ln(y/x) \)

directly derived from \( W(y)=x \) iff \( xe^x=y \) by making the iteration formula \( x=\ln(y/x) \) out of it. The branches of the logarithm then again correspond to the branches of \( W \).

For example for \( y=-\frac{\pi}{2} \) I get by this iteration formula: \( W(-\frac{\pi}{2})=1.5708 i\approx \frac{\pi}{2}i \) when using the main branch of the logarithm.