Cauchy principal value

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

• the finite number

<math>\lim_{\varepsilon\rightarrow 0+} \left(\int_a^{b-\varepsilon} f(x)\,dx+\int_{b+\varepsilon}^c f(x)\,dx\right)<math>

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

<math>\int_a^b f(x)\,dx=\pm\infty<math>

for any a < b and

<math>\int_b^c f(x)\,dx=\mp\infty<math>

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

or

• the finite number

<math>\lim_{a\rightarrow\infty}\int_{-a}^a f(x)\,dx<math>

where

<math>\int_{-\infty}^0 f(x)\,dx=\pm\infty<math>

and

<math>\int_0^\infty f(x)\,dx=\mp\infty<math>

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

Examples

Consider the difference in values of two limits:

<math>\lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_a^1\frac{dx}{x}\right)=0,<math>

<math>\lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_{2a}^1\frac{dx}{x}\right)=-\log_e 2.<math>

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

<math>\int_{-1}^1\frac{dx}{x}{\ }

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

Similarly, we have

<math>\lim_{a\rightarrow\infty}\int_{-a}^a\frac{2x\,dx}{x^2+1}=0,<math>

but

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

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

<math>\int_{-\infty}^\infty\frac{2x\,dx}{x^2+1}{\ }

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

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