blob: 02a8ecea0e9501f960e72fa5589eae5abf223ef5 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
|
---
title: "Residue theorem"
firstLetter: "R"
publishDate: 2021-11-13
categories:
- Mathematics
- Complex analysis
date: 2021-11-13T20:51:13+01:00
draft: false
markup: pandoc
---
# Residue theorem
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).
|
