Cauchy principal value

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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 "−").

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

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

Contents

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 \int f(x)\,dx, P, P.V., \mathcal{P}, Pv, (CPV) and V.P..

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)=-\ln 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}=-\ln 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.

Let C_0^\infty(\mathbb{R}) be the set of smooth functions with compact support on the real line \mathbb{R}. Then, the map

\operatorname{p.\!v.}\left(\frac{1}{x}\right)\,: C_0^\infty(\mathbb{R}) \to \mathbb{C}

defined via the Cauchy principal value as

\operatorname{p.\!v.}\left(\frac{1}{x}\right)(u)=\lim_{\varepsilon\to 0+} \int_{| x|>\varepsilon} \frac{u(x)}{x} \, dx for u\in C_0^\infty(\mathbb{R})

is a distribution.

This article incorporates material from Cauchy principal part integral on PlanetMath, which is licensed under the GFDL.

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