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---
title: "Cauchy principal value"
sort_title: "Cauchy principal value"
date: 2021-11-01
categories:
- Mathematics
layout: "concept"
---
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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