Let $f$ be an analytic function in two complex variables. It is well known that we can expand $f$ in a convergent series of two variables.
Can we separate the variables in such a manner that $f$ becomes a product of two entire functions in one variable?
This is not possible in general. For example, take $f(z, w) = z + w$ and suppose $g(z)h(w) = z + w$. Then $g(0)h(w) = w$ (so $g(0) \neq 0$) and $g(z)h(0) = z$ (so $h(0) \neq 0$). But we also have $g(0)h(0) = 0$ which is a contradiction as $g(0) \neq 0$ and $h(0) \neq 0$.