Note that u 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 g(z;k)≡f(z)+u(|f(z)|−k−1)kSin(z). This will yield the same roots as f(z)=x3−x on the real line. This obtains from Rouché's theorem as long as |f|<k, sin(z) being of bounded modulus there.
Elsewhere the value can be seen as depending on the hyperbolic sine of the imaginary part of z, sinh(ℑz). Proof of which.
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