Cauchy principal value

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

• the finite number

$\lim _{\varepsilon \rightarrow 0+}\left(\int _{a}^{b-\varepsilon }f(x)\,dx+\int _{b+\varepsilon }^{c}f(x)\,dx\right)$

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

$\int _{a}^{b}f(x)\,dx=\pm \infty$

for any a < b and

$\int _{b}^{c}f(x)\,dx=\mp \infty$

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

or

• the finite number

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

where

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

and

$\int _{0}^{\infty }f(x)\,dx=\mp \infty$

(again, one sign is "+" and the other is "−").

Examples

Consider the difference in values of two limits:

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

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

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

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

Similarly, we have

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

but

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

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

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

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