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:


 * $\displaystyle \operatorname{Res} \left({f, a}\right) = \lim_{z \mathop \to a} \left({z - a}\right) f \left({z}\right)$

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


 * $\displaystyle f\left({z}\right) = \sum_{n \mathop = -\infty}^\infty c_n \left({z - a}\right)^n$

convergent in $D \setminus \left\{a\right\}$, where $\left({c_n}\right)$ is a doubly infinite sequence of complex coefficients.

As $f$ has only a simple pole at $a$, $c_n = 0$ for $n < -1$. So, we can write:


 * $\displaystyle f\left({z}\right) = \sum_{n \mathop = 0}^\infty c_n \left({z - a}\right)^n + \frac { c_{-1} } {z - a}$

Then: