Define the reals $x,y$ as
$$ y = \max \Im ( W ( - exp(x)) exp(-x) ) $$
Where $W$ is the standard Lambert W function and $\Im$ means the imaginary part.
How to find $x$ and $y$ ?
Closed forms ( allowing integrals , sums etc ) , contour integrals , numerical methods ??
I know how to express the local COMPLEX max on the complex plane for an analytic function by a contour integral.
I also know the Cauchy-Riemann equations that related the real and imaginary parts of an analytic function by differential equations.
Yet this does not appear to help me. Maybe it should help me , but i do not see how.
I ask here for a case of the $W$ function , because i do not want to ask too General questions. But i am also intrested in General methods ofcourse.