Suppose I have an expression of the form
$$f(z) := f_1(z)+f_2(z)$$ ($f,f_1,f_2$ can e.g. be integrals) with $f_1$ convergent in the region $R_1=\{\Re(z)>-1\}$ and $f_2$ convergent in the region $R_2=\{\Re(z)<1\}$.
Let $R= R_1\cap R_2$. Then, to my knowledge, $f$ is convergent in $R$. My question is when is it possible to meromorphically extend $f$ to a larger region, say all of $\mathbb{C}$?