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

• the finite number

$\backslash lim\_\{\backslash varepsilon\backslash rightarrow\; 0+\}\; \backslash leftf(x)\backslash ,dx+\backslash int\_\{b+\backslash varepsilon\}^c\; f(x)\backslash ,dx\backslash right$

where b is a point at which the behavior of the function f is such that

$\backslash int\_a^b\; f(x)\backslash ,dx=\backslash pm\backslash infty$

for any a < b and

$\backslash int\_b^c\; f(x)\backslash ,dx=\backslash mp\backslash infty$

for any c > b (one sign is "+" and the other is "−").

or

• the finite number

$\backslash lim\_\{a\backslash rightarrow\backslash infty\}\backslash int\_\{-a\}^a\; f(x)\backslash ,dx$

where

$\backslash int\_\{-\backslash infty\}^0\; f(x)\backslash ,dx=\backslash pm\backslash infty$

and

$\backslash int\_0^\backslash infty\; f(x)\backslash ,dx=\backslash mp\backslash infty$

(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

$\backslash lim\_\{\backslash varepsilon\; \backslash rightarrow\; 0+\}\backslash int\_\{b-\backslash frac\{1\}\{\backslash varepsilon\}\}^\{b-\backslash varepsilon\}\; f(x)\backslash ,dx+\backslash int\_\{b+\backslash varepsilon\}^\{b+\backslash frac\{1\}\{\backslash varepsilon\}\}f(x)\backslash ,dx.$

Nomenclature

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

Examples

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Consider the difference in values of two limits:

$\backslash lim\_\{a\backslash rightarrow\; 0+\}\backslash left(\backslash int\_\{-1\}^\{-a\}\backslash frac\{dx\}\{x\}+\backslash int\_a^1\backslash frac\{dx\}\{x\}\backslash right)=0,$

$\backslash lim\_\{a\backslash rightarrow\; 0+\}\backslash left(\backslash int\_\{-1\}^\{-a\}\backslash frac\{dx\}\{x\}+\backslash int\_\{2a\}^1\backslash frac\{dx\}\{x\}\backslash right)=-\backslash log\_e\; 2.$

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

$\backslash int\_\{-1\}^1\backslash frac\{dx\}\{x\}\{\backslash \; \}$

\left(\mbox{which}\ \mbox{gives}\ -\infty+\infty\right).

Similarly, we have

$\backslash lim\_\{a\backslash rightarrow\backslash infty\}\backslash int\_\{-a\}^a\backslash frac\{2x\backslash ,dx\}\{x^2+1\}=0,$

but

$\backslash lim\_\{a\backslash rightarrow\backslash infty\}\backslash int\_\{-2a\}^a\backslash frac\{2x\backslash ,dx\}\{x^2+1\}=-\backslash log\_e\; 4.$

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

$\backslash int\_\{-\backslash infty\}^\backslash infty\backslash frac\{2x\backslash ,dx\}\{x^2+1\}\{\backslash \; \}$

\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

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Let $C\_0^\backslash infty(\backslash mathbb\{R\})$ be the set of smooth functions with compact support on the real line $\backslash mathbb\{R\}.$ Then, the map

$\backslash operatorname\{p.\backslash !v.\}\backslash left(\backslash frac\{1\}\{x\}\backslash right)\backslash ,:\; C\_0^\backslash infty(\backslash mathbb\{R\})\; \backslash to\; \backslash mathbb\{C\}$

defined via the Cauchy principal value as

$\backslash operatorname\{p.\backslash !v.\}\backslash left(\backslash frac\{1\}\{x\}\backslash right)(u)=\backslash lim\_\{\backslash varepsilon\backslash to\; 0+\}\; \backslash int\_\{|\; x|>\backslash varepsilon\}\; \backslash frac\{u(x)\}\{x\}\; \backslash ,\; dx$ for $u\backslash in\; C\_0^\backslash infty(\backslash mathbb\{R\})$

is a distribution.

Mathematical analysis

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Cauchyscher Hauptwert | Koŝia ĉefa valoro | コーシーの主値