Suppose I have the following formula, \begin{equation} \mathrm{P}\int_{-\infty}^{\infty} f(x,q) dx = F(q), \end{equation} for all $q\in\mathbb{R}$, where P stands for the Cauchy principal value.
If $F(q)$ is analytic for all $q\in\mathbb{C}$, and if $\int_{-\infty}^{\infty} f(x,q) dx$ exists for all $q$ with $\Im[q]\not=0$, then does the formula, \begin{equation} \int_{-\infty}^{\infty} f(x,q) dx = F(q), \end{equation} hold for all $q\in\mathbb{C}$ such that $\Im[q]\not=0$?
If I need to be more specific, I have the following type of integrand in my mind: \begin{equation} f(x,q) = \frac{g(x)}{x-q}, \end{equation} where $g(x)$ is analytic for all $x\in \mathbb{C}$.
In other words, is it justified to analytically continue an integral representation of a function if the integral is originally given as the Cauchy principal value?