# Cauchy principal value

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

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

or

• the finite number
[itex]\lim_{a\rightarrow\infty}\int_{-a}^a f(x)\,dx[itex]
where
[itex]\int_{-\infty}^0 f(x)\,dx=\pm\infty[itex]
and
[itex]\int_0^\infty f(x)\,dx=\mp\infty[itex]
(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

[itex]\lim_{\varepsilon \rightarrow 0+}\int_{b-1/\varepsilon}^{b-\varepsilon} f(x)\,dx+\int_{b+\varepsilon}^{b+1/\varepsilon}f(x)\,dx.[itex]

## Nomenclature

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

## Examples

Consider the difference in values of two limits:

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

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

[itex]\int_{-1}^1\frac{dx}{x}{\ }

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

Similarly, we have

[itex]\lim_{a\rightarrow\infty}\int_{-a}^a\frac{2x\,dx}{x^2+1}=0,[itex]

but

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

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

[itex]\int_{-\infty}^\infty\frac{2x\,dx}{x^2+1}{\ }

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

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

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