Mittag-Leffler's Expansion Theorem

Theorem
Let $f$ be a meromorphic function with only simple poles continuous, or with a removable singularity, at $0$.

Let $X$ be the set of poles of $f$.

For $N \in \N$, let $C_N$ be a circle, centred at the origin, of radius $R_N$, where $R_N \to \infty$ as $N \to \infty$, such that $\partial C_N$ contains no poles of $f$ for any $N$.

Let $M > 0$ be a real number independent of $N$ such that for all $z \in \partial C_N$, $\cmod {\map f z} < M$, for all $N \in \N$.

Then:
 * $\displaystyle \map f z = \map f 0 + \sum_{n \mathop \in X} \Res f n \paren {\frac 1 {z - n} + \frac 1 n}$

where:
 * $\Res f n$ is the residue of $f$ at $n$
 * $z$ is not a pole of $f$.