I would like to know if there are any methods to find the roots (analytically) of complex valued equations of the following form:
$$ f(z)=P(z)+\frac{e^{-z}}{(1+e^{-z})^2} $$
where $P(z)$ is a complex valued polynomial with real coefficients and the latter term is the derivative with respect to the standard first-order logistic equation.
Any suggestions, hints, or recommended readings would be greatly appreciated.