Zeros of functions analytic in 2 variables

[Paul Colby]

TL;DR

Let, $f(z_1,z_2)$, be entitre analytic in both $z_1$ and $z_2$. If $f(z_1,z_2)=0$ has a solution, is it true that $f$ is zero along some curve in $z_1,z_2$?

This question crops up in solving electromagnetic boundary value problems. For problems with rotational symmetry, if $f$ has a node at $(\theta,\phi)$ then $(\theta,\phi')=0$ for all other $\phi'$. This (I think) implies that, $$f(\theta,\phi)=F(\theta)G(\phi)$$ which, for the problems I'm considering, isn't going to happen except under very special circumstances.

My thinking is, if a node exists at $(z_1,z_2)$, then some curve in $(z_1,z_2)$ must exist along which $f$ is zero.

 

[Paul Colby]

Ah, problem solved. In the form asked, one simply solves $f(z_1,z_2)=0$ for $z_2$ in terms of $z_1$. The real problem that was troubling me is addressed by realizing real polar coordinates, $(\theta,\phi)$, are in fact just a single complex coordinate. The functions, $f$, are really entire functions of a single complex coordinate. The zeros of analytic functions that are not identically zero, are all isolated points.