Consider a function $F(x, y)$ of two complex variables. For $\Re(y)>0$, we know the analytic structure of the function. In that case, the function is meromorphic, with simple poles in $x$ at locations $x=w_i$, for $i=0, 1, ...$ . The locations of the poles do not depend on $y$, however the residues do depend on $y$.

Furthermore, we know that $F(x, y)$ is symmetric in $x$ and $y$. So, we also know its analytic structure for $\Re(x)>0$. In that case, the function has poles in $y$ at $y=w_i$. The problem is to know the analytic structure of the function when $\Re(x)<0$ and $\Re(y)<0$ simultaneously. I suspect that, in that case, the function will still be meromorphic, with poles at $x=w_i$ and $y=w_j$.

I came across this problem, in the context of physics. I am a theoretical physics PhD student, so I do not even know how to start. If somebody knows the solution to the problem, or has any helpful comments, I would be very grateful.