---
title: The Calculus of Residues
parent: One Complex Variable
grand_parent: Complex Analysis
nav_order: 4
---

# {{ page.title }}

{% definition Residue %}
TODO
{% enddefinition %}

Calculation of Residues

If $f$ has a simple pole at $c$, then
$\Res(f,c) = \lim_{z \to c} (z-c) f(z)$.

If $f$ has a pole of order $k$ at $c$, then

$$
\Res(f,c) = \frac{1}{(k-1)!} g^{(k-1)}(c), \quad \text{where} \ g(z) = (z-c)^k f(z).
$$

If $g$ and $h$ are holomorphic near $c$ and $h$ has a simple zero at $c$,
then $f = g/h$ has a simple pole at $c$ and

$$
\Res(f,c) = \frac{g(c)}{h'(c)}
$$

{% theorem * Residue Theorem (Basic Version) %}
Let $f$ be a function meromorphic in an open subset $G \subset \CC$.
Let $\gamma$ be a contour in $G$ such that
the interior of $\gamma$ is contained in $G$
and contains finitely many poles $c_1, \ldots, c_n$ of $f$.
Then

$$
\int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k)
$$
{% endtheorem %}

{% proof %}
{% endproof %}

TODO
- argument principle
- Rouché's theorem
- winding number