Convergence of residues of meromorphic functions

Question

Let $\Omega$ be a domain in $\mathbb{C}$ and $z_0 \in \Omega$. Further, let $f_n$ and $f$ be holomorphic functions on $\Omega \setminus \{z_0\}$ such that $z_0$ is a simple pole for each $f_n$ and at most a simple pole (i.e. a simple pole or a removable singularity) for $f$. If we assume that $f_n \to f$ uniformly on any compact subset of $\Omega \setminus \{z_0\}$, is there anything we can say about the behaviour of the Residues $\operatorname{Res}_{z_0}(f_n)$ under taking the limit?

Answer 1

The Residue Theorem is usually used to evaluate integrals in terms of residues, but here you'll use it to write a residue as an integral (let's say around a circle around $z_0$ within $\Omega$). The circle is a compact set, so $f_n$ converges uniformly to $f$ on it. Therefore...

Answer 2

Let$$g_n(z)=\begin{cases}zf_n(z)&\text{ if }z\neq z_0\\\lim_{z\to z0}zf_n(z)&\text{ if }z=z_0.\end{cases}$$Then $(g_n)_{n\in\mathbb N}$ is a sequence of holomorphic functions that converges to$$\begin{array}{rccc}g\colon&\Omega\cup\{z_0\}&\longrightarrow&\mathbb C\\&z&\mapsto&\begin{cases}zf(z)&\text{ if }z\neq z_0\\\lim_{z\to z0}zf(z)&\text{ if }z=z_0\end{cases}\end{array}$$uniformly on any compact subset of $\Omega\cup\{z_0\}$. In particular, $g(z_0)=\lim_{n\in\mathbb N}g_n(z_0)$. But this means that$$\operatorname{Res}_{z_0}(f)=\lim_{n\in\mathbb N}\operatorname{Res}_{z_0}(f_n).$$

Answer 3

I can't quite comment yet, so I'll leave you my hint here: Recall the integral definition of the residue and the dominated convergence theorem. I'm sure you can take it from there :)