---
title: "Residue theorem"
sort_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 nonzero
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 $$z_p$$:

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

**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 $$C$$:

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


{% include proof/start.html id="proof-theorem" -%}
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:

$$\begin{aligned}
    \oint_C f(z) \dd{z}
    &= \oint_C h(z) \dd{z} + \sum_{p} R_p \oint_C \frac{1}{z - z_p} \dd{z}
    = \sum_{p} R_p \: 2 \pi i
\end{aligned}$$
{% include proof/end.html id="proof-theorem" %}


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).