Here is a slightly more interesting output
Note that here is the Heaviside step function but it's applied to the (shifted) modulus of a function rather than a variable. That is, we are interested in roots of . This will yield the same roots as on the real line. This obtains from Rouché's theorem as long as , being of bounded modulus there. Elsewhere the value can be seen as depending on the hyperbolic sine of the imaginary part of , . Proof of which.