If $f(z)$ is an analytic function in a complex neighborhood (in $\mathbb C^n$) of a real point $x^0 \in \mathbb R^n \subset \mathbb C^n$, then $\log|f(x)|$ is integrable over some neighborhood $U \subset \mathbb R^n$ of $x^0$. Does anyone know a reference for this fact? I looked into a number of books on several complex variables without finding the statement.
I want a reference, because I don't think the validity of the assertion is obvious. However, here is sketch of proof. We may assume that $x^0= (0, \ldots, 0)$. Choose coordinates so that $f(0, \ldots 0, x_n)$ does not vanish identically near $x_n= 0$. Then the same is true for $x'$ in some neighborhood $V$ of $(0, \ldots, 0) \in \mathbb R^{n-1}$, hence $x_n \mapsto \log|f(x', x_n)|$ is locally integrable for every $x' \in V$. But we need to prove that the integrability is uniform with respect to $x'$. However, using Weierstrass' preparation theorem it is now easy to see that the integral $\int\log|f(x', x_n)|dx_n$ over a neighborhood of $x_n = 0$ is locally bounded as a function of $x'$ after (if needed) a choice of new coordinates.
