Let $f(x):\mathbf{R} \rightarrow \mathbf{R}$ be a real function which extends meromorphically to the complex $\mathbf{C}$ plane to a function $\tilde f(z) : \mathbf{C} \rightarrow \mathbf{C}$. Let then $f_n(x): \mathbf{R} \rightarrow \mathbf{R}$, with again meromorphic extensions $\tilde f_n(x): \mathbf{C} \rightarrow \mathbf{C}$, such that:
$f_n(x) \rightarrow f(x), x \in \mathbf{R}$
with some convergence criterion (for example uniformly or in some normed sense). Is it true that also:
$\tilde f_n(z) \rightarrow \tilde f(z), z \in \mathbf{C}$
for some convergence criterion???
For example can we say that the set of poles of $\tilde f_n$ will tend to the poles of $\tilde f$
I came across this question when trying to understand how analytical continuations can be practically performed. I did not find an answer yet.
Define, for each $n\in\mathbb N$,$$\begin{array}{rccc}f_n\colon&\mathbb{C}\setminus\left\{\pm\frac in\right\}&\longrightarrow&\mathbb C\\&z&\mapsto&\displaystyle\frac{\sin\bigl(\sqrt nz\bigr)}{1+n^2z^2}.\end{array}$$Then the sequence $\bigl(f_n|_{\mathbb R}\bigr)_{n\in\mathbb N}$ converges (uniformly, I think, but I am not entirely sure) to the null function. However, there is no $z\in\mathbb{C}\setminus\mathbb R$ such that the sequence $\bigl(f_n(z)\bigr)_{n\in\mathbb N}$ converges.