---
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.