I have a iterative process for applying steepest descent/saddle point approximation to functions of several complex variables. It only really works if the function is separable, though. the function in question $f:\mathbb{C}^2 \to \mathbb{C}$ is holomorphic in both variables on the domain (it's actually entire for the current function I'm working with, I'm not sure if this would make a difference).
Is there some way to test if it's possible to test for separability, or make some deformation/transform to enable this?
Formally, I want to define separability roughly as follows: for $f:\mathbb{C}^2\to\mathbb{C}$, f separable if $$f(x,y)=g(x)h(y)$$ Where $g,h$ are not constant (unless $\frac{d}{dx} f=0/\frac{d}{dy}f=0$ on the domain )
Further question: so, maybe a function will not be separable in x,y. However, is there a way to construct/determine constructability of $u,v=u(x,y),v(x,y)$ such that $f(u,v) $ is separable?