# Cauchy principal value

In mathematics, the Cauchy principal value of certain improper integrals is defined as either

• the finite number
$\displaystyle \lim_{\varepsilon\rightarrow 0+} \left[\int_a^{b-\varepsilon} f(x)\,dx+\int_{b+\varepsilon}^c f(x)\,dx\right]$
where b is a point at which the behavior of the function f is such that
$\displaystyle \int_a^b f(x)\,dx=\pm\infty$
for any a < b and
$\displaystyle \int_b^c f(x)\,dx=\mp\infty$
for any c > b (one sign is "+" and the other is "−").

or

• the finite number
$\displaystyle \lim_{a\rightarrow\infty}\int_{-a}^a f(x)\,dx$
where
$\displaystyle \int_{-\infty}^0 f(x)\,dx=\pm\infty$
and
$\displaystyle \int_0^\infty f(x)\,dx=\mp\infty$
(again, one sign is "+" and the other is "−").

In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity. This is usually done by a limit of the form

$\displaystyle \lim_{\varepsilon \rightarrow 0+}\int_{b-\frac{1}{\varepsilon}}^{b-\varepsilon} f(x)\,dx+\int_{b+\varepsilon}^{b+\frac{1}{\varepsilon}}f(x)\,dx.$

## Nomenclature

The Cauchy principal value of a function $\displaystyle f$ can take on several nomenclatures, varying for different authors. These include (but are not limited to): $\displaystyle PV \int f(x)\,dx$ , $\displaystyle P$ , P.V., $\displaystyle \mathcal{P}$ , $\displaystyle P_v$ , $\displaystyle (CPV)$ and V.P..

## Examples

Consider the difference in values of two limits:

$\displaystyle \lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_a^1\frac{dx}{x}\right)=0,$
$\displaystyle \lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_{2a}^1\frac{dx}{x}\right)=-\log_e 2.$

The former is the Cauchy principal value of the otherwise ill-defined expression

$\displaystyle \int_{-1}^1\frac{dx}{x}{\ } \left(\mbox{which}\ \mbox{gives}\ -\infty+\infty\right).$

Similarly, we have

$\displaystyle \lim_{a\rightarrow\infty}\int_{-a}^a\frac{2x\,dx}{x^2+1}=0,$

but

$\displaystyle \lim_{a\rightarrow\infty}\int_{-2a}^a\frac{2x\,dx}{x^2+1}=-\log_e 4.$

The former is the principal value of the otherwise ill-defined expression

$\displaystyle \int_{-\infty}^\infty\frac{2x\,dx}{x^2+1}{\ } \left(\mbox{which}\ \mbox{gives}\ -\infty+\infty\right).$

These pathologies do not afflict Lebesgue-integrable functions, that is, functions the integrals of whose absolute values are finite.

## Distribution theory

Let $\displaystyle C_0^\infty(\mathbb{R})$ be the set of smooth functions with compact support on the real line $\displaystyle \mathbb{R}.$ Then, the map

$\displaystyle \operatorname{p.\!v.}\left(\frac{1}{x}\right)\,: C_0^\infty(\mathbb{R}) \to \mathbb{C}$

defined via the Cauchy principal value as

$\displaystyle \operatorname{p.\!v.}\left(\frac{1}{x}\right)(u)=\lim_{\varepsilon\to 0+} \int_{| x|>\varepsilon} \frac{u(x)}{x} \, dx$ for $\displaystyle u\in C_0^\infty(\mathbb{R})$

is a distribution.