Analytic continuation of 2 variable function

Question

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.

Answer 1

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.

The proof, essentially by a slightly clever use of Cauchy's theorem in one variable, is reproduced various places. Langlands' SLN 544 is on-line nowadays, for example. The lemma is in various on-line notes of mine, e.g., appendix 3A in my Cambridge Univ. Press book "Modern Analysis of Automorphic Forms by Example". This is also (legally!) on-line linked-to from http://www.math.umn.edu/~garrett/m/v/