---
title: "Residue theorem"
date: 2021-11-13
categories:
- Mathematics
- Complex analysis
layout: "concept"
---

A function $f(z)$ is **meromorphic** if it is
[holomorphic](/know/concept/holomorphic-function/)
except in a finite number of **simple poles**,
which are points $z_p$ where $f(z_p)$ diverges,
but where the product $(z - z_p) f(z)$ is non-zero
and still holomorphic close to $z_p$.
In other words, $f(z)$ can be approximated close to $z_p$:

$$\begin{aligned}
    f(z)
    \approx \frac{R_p}{z - z_p}
\end{aligned}$$

Where the **residue** $R_p$ of a simple pole $z_p$ is defined as follows, and
represents the rate at which $f(z)$ diverges close to $z_p$:

$$\begin{aligned}
    \boxed{
        R_p = \lim_{z \to z_p} (z - z_p) f(z)
    }
\end{aligned}$$

**Cauchy's residue theorem** for meromorphic functions
is a generalization of Cauchy's integral theorem for holomorphic functions,
and states that the integral on a contour $C$
purely depends on the simple poles $z_p$ enclosed by $C$:

$$\begin{aligned}
    \boxed{
        \oint_C f(z) \dd{z} = i 2 \pi \sum_{z_p} R_p
    }
\end{aligned}$$

<div class="accordion">
<input type="checkbox" id="proof-res-theorem"/>
<label for="proof-res-theorem">Proof</label>
<div class="hidden">
<label for="proof-res-theorem">Proof.</label>
From the definition of a meromorphic function,
we know that we can decompose $f(z)$ like so,
where $h(z)$ is holomorphic and $z_p$ are all its poles:

$$\begin{aligned}
    f(z) = h(z) + \sum_{z_p} \frac{R_p}{z - z_p}
\end{aligned}$$

We integrate this over a contour $C$ which contains all poles, and apply
both Cauchy's integral theorem and Cauchy's integral formula to get:

$$\begin{aligned}
    \oint_C f(z) \dd{z}
    &= \oint_C h(z) \dd{z} + \sum_{p} R_p \oint_C \frac{1}{z - z_p} \dd{z}
    = \sum_{p} R_p \: 2 \pi i
\end{aligned}$$
</div>
</div>

This theorem might not seem very useful,
but in fact, by cleverly choosing the contour $C$,
it lets us evaluate many integrals along the real axis,
most notably [Fourier transforms](/know/concept/fourier-transform/).
It can also be used to derive the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations).