Cauchy principal value

From Academic Kids

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 "−").

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

<math>\lim_{\varepsilon \rightarrow 0+}\int_{b-1/\varepsilon}^{b-\varepsilon} f(x)\,dx+\int_{b+\varepsilon}^{b+1/\varepsilon}f(x)\,dx.<math>

Nomenclature

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

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.

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