Suppose I define a function using an integral:
$f(z)=\int_{\mathbb R} g(z,x)\ dx,$
where $g$ is some function, $z$ is a complex variable, and $x$ is a real variable. Suppose the integral exists for $z\in U$, where $U$ is some open region. What are sufficient conditions on $g$ so that $f$ is analytic here, and why do they suffice?
I looked in Ahlfors but couldn't find anything relevant.