Residue at Simple Pole

Theorem
Let $f: \C \to \C$ be a function meromorphic on some region, $D$, containing $a$.

Let $f$ have a simple pole at $a$.

Then the residue of $f$ at $a$ is given by:


 * $\ds \Res f a = \lim_{z \mathop \to a} \paren {z - a} \map f z$

Proof
By Existence of Laurent Series, there exists a Laurent series:


 * $\ds \map f z = \sum_{n \mathop = -\infty}^\infty c_n \paren {z - a}^n$

which is convergent in $D \setminus \set a$, where $\sequence {c_n}$ is a doubly infinite sequence of complex coefficients.

We are given that $f$ has only a simple pole at $a$.

Thus $c_n = 0$ for $n < -1$.

So we can write:


 * $\ds \map f z = \sum_{n \mathop = 0}^\infty c_n \paren {z - a}^n + \frac {c_{-1} } {z - a}$

Then: