Does an analytic function $f(x|y)$, $x,y \in C$ holds its analyticity after conditioning on a random variable $y$ e.g. $g(x,y)=E[f(x|y)]$?

Question

Let I have a function $f(x|y)$ is holomorphic (and Analytic in the complex plane) where $y$ is deterministic and $x,y\in C$.

In this part $y$ is no more determinstic. $h$ is a random variable; both real part $ \Re (y)$ and imaginary part $\Im (y)$ is gaussian distributed random variable with mean $\mu$ and variance $\sigma^2$. Now let define a new function $g(x,y)$ such that,

$g(x,y)=E[f(x|y)]$

Where $E[.]$ is the expectation operation. My question is

1. Is $g(x,y)$ analytic on the entire complex plane of $X$?

Can I prove it? Any hints or suggestions will be helpful (I am new in analysis).