# Analytic continuation of 2 variable function

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.

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There is a related idea sometimes known as "Bochner's Lemma" (I first saw this as part of an appendix in R. Langlands' Springer Lecture Notes in Math 544). Let $$f(z,w)$$ be a holomorphic function of two complex variables $$z,w$$ (I feel more comfortable having complex variables $$z,w$$ rather than $$x,y$$) in the region where $$\Re(z)>0$$ or $$\Re(w)>0$$, and with "vertical" growth bounded by $$|f(z,w)|\le C\cdot e^{(1+|\Im(z)|+|\Im(w))^\alpha}$$ for some real $$\alpha<2$$. Then $$f(z,w)$$ has an analytic continuation to all of $$\mathbb C^2$$.
More generally, for a connected region $$\Omega\subset\mathbb R^2$$, holomorphic $$f$$ on the "tube domain" $$(\Re(z),\Re(w))\in \Omega$$, with the same vertical growth bound, extends to the convex closure of that region.