From b090363af28c577bbf9da60d03c82056036588aa Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 1 Nov 2021 21:29:02 +0100
Subject: Expand knowledge base

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+---
+title: "Cauchy principal value"
+firstLetter: "C"
+publishDate: 2021-11-01
+categories:
+- Mathematics
+
+date: 2021-11-01T12:54:50+01:00
+draft: false
+markup: pandoc
+---
+
+# Cauchy principal value
+
+The **Cauchy principal value** $\mathcal{P}$,
+or just **principal value**,
+is a method for integrating problematic functions,
+i.e. functions with singularities,
+whose integrals would otherwise diverge.
+
+Consider a function $f(x)$ with a singularity at some finite $x = b$,
+which is hampering attempts at integrating it.
+To resolve this, we define the Cauchy principal value $\mathcal{P}$ as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \mathcal{P} \int_a^c f(x) \dd{x}
+        = \lim_{\varepsilon \to 0^{+}} \!\bigg( \int_a^{b - \varepsilon} f(x) \dd{x} + \int_{b + \varepsilon}^c f(x) \dd{x} \bigg)
+    }
+\end{aligned}$$
+
+If $f(x)$ instead has a singularity at postive infinity $+\infty$,
+then we define $\mathcal{P}$ as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \mathcal{P} \int_{a}^\infty f(x) \dd{x}
+        = \lim_{c \to \infty} \!\bigg( \int_{a}^c f(x) \dd{x} \bigg)
+    }
+\end{aligned}$$
+
+And analogously for $-\infty$.
+If $f(x)$ has singularities both at $+\infty$ and at $b$,
+then we simply combine the two previous cases,
+such that $\mathcal{P}$ is given by:
+
+$$\begin{aligned}
+    \mathcal{P} \int_{a}^\infty f(x) \:dx
+    = \lim_{c \to \infty} \lim_{\varepsilon \to 0^{+}}
+    \!\bigg( \int_{a}^{b - \varepsilon} f(x) \:dx + \int_{b + \varepsilon}^{c} f(x) \:dx \bigg)
+\end{aligned}$$
+
+And so on, until all problematic singularities have been dealt with.
+
+In some situations, for example involving
+the [Sokhotski-Plemelj theorem](/know/concept/sokhotski-plemelj-theorem/),
+the symbol $\mathcal{P}$ is written without an integral,
+in which case the calculations are implicitly integrated.
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