Cauchy principal value: Definition from Answers.com

This article is about a method for assigning values to improper integrals. For the values of a complex function associated with a single branch, see Principal value. For the negative-power portion of a Laurent series, see Principal part.

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

1 Formulation
2 Examples
3 Distribution theory
4 Nomenclature
5 See also
6 References and notes

Formulation

Depending on the type of singularity in the integral, the Cauchy principal value is defined as one of the following:

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 "−"; see plus or minus for precise usage of notations ±, ∓).

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, see plus or minus for precise usage of notation ±, ∓).

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

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

or

in terms of contour integrals of a complex-valued function f(z); z = x + iy, with a pole on the contour. The pole is enclosed with a circle of radius ε and the portion of the path outside this circle is denoted L(ε). Provided the function f(z) is integrable over L(ε) no matter how small ε becomes, then the Cauchy principal value is the limit:^[1]

$\mathrm{P} \int_{L} f(z) \ dz = \int_L^* f(z)\ dz = \lim_{\varepsilon \to 0 } \int_{L( \varepsilon)} f(z)\ dz \ ,$

where two of the common notations for the Cauchy principal value appear on the left of this equation.

Examples

Consider the difference in values of two limits:

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

$\lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_{2a}^1\frac{dx}{x}\right)=-\ln 2.$

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.

Distribution theory

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

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

is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears for example in the Fourier transform of the Heaviside step function.

More generally, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space Rⁿ. If K(x) has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal value distribution is defined on compactly supported smooth functions by

$(p.v. K)(f) = \lim_{\epsilon\to 0}\int_{\mathbb{R}^n\setminus B_\epsilon(0)} f(x)K(x)\,dx.$

Such a limit may not be well defined or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if K is a continuous homogeneous function of degree −n whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.

Nomenclature

The Cauchy principal value of a function $f$ can take on several nomenclatures, varying for different authors. Among these are:

$PV \int f(x)\,dx,\quad \int_L^* f(z)\, dz,\quad -\!\!\!\!\!\!\int f(x)\,dx,$ $P\ ,$ P.V. $, \mathcal{P}\ ,$ $P_v\ ,$ $(CPV)\ ,$ and V.P.

References and notes

^ Ram P. Kanwal (1996). Linear Integral Equations: theory and technique (2nd Edition ed.). Boston: Birkhäuser. p. 191. ISBN 0817639403. http://books.google.com/books?id=-bV9Qn8NpCYC&pg=PA194&lpg=PA194&dq=+%22Poincar%C3%A9-Bertrand+transformation%22&source=web&ots=iofB7oQccG&sig=2yieQ-eUpZTZtPcZrJJpBZAO-R4&hl=en#PPA191,M1.

Please help improve this article by expanding it. Further information might be found on the talk page. (October 2008)

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Cauchy principal value

Contents

Formulation

Examples

Distribution theory

Nomenclature

See also

References and notes