From c0d352dd0f66b47ee91fb96eaf320f895fa78790 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 14 Nov 2021 17:54:04 +0100
Subject: Expand knowledge base

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+---
+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).
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