---
title: The Calculus of Residues
parent: One Complex Variable
grand_parent: Complex Analysis
nav_order: 4
# cspell:words
#published: false
---

# {{ page.title }}

{: .definition-title }
> Definition (Residue)
>
> TODO

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-title }
> Residue Theorem (Basic Version)
> {: #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
>
> 
> $$
> \int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k)
> $$
> {: .katex-display .mb-0 }

{% proof %}
{% endproof %}

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