Cauchy principal part integral

Definition [1, 2, 3] Let $C_{0}^{\infty}(\mathbb{R})$ be the set of smooth functions with compact support on $\mathbb{R}$ . Then the Cauchy principal part integral (or, more in line with the notation, the Cauchy principal value) $\operatorname{p.\!v.}(\frac{1}{x})$ is mapping $\operatorname{p.\!v.}(\frac{1}{x})\,:C_{0}^{\infty}(\mathbb{R})\to\mathbb{C}$ defined as

\operatorname{p.\!v.}(\frac{1}{x})(u)=\lim_{\varepsilon\to 0+}\int_{|x|>% \varepsilon}\frac{u(x)}{x}dx

for $u\in C_{0}^{\infty}(\mathbb{R})$ .

Theorem The mapping $\operatorname{p.\!v.}(\frac{1}{x})$ is a distribution of first order (http://planetmath.org/Distribution4). That is, $\operatorname{p.\!v.}(\frac{1}{x})\in\mathcal{D}^{\prime 1}(\mathbb{R})$ .

(proof. (http://planetmath.org/Operatornamepvfrac1xIsADistributionOfFirstOrder))

0.0.1 Properties

1.

The distribution $\operatorname{p.\!v.}(\frac{1}{x})$ is obtained as the limit ([3], pp. 250)

$\frac{\chi_{n|x|}}{x}\to\operatorname{p.\!v.}(\frac{1}{x}).$

as $n\to\infty$ . Here, $\chi$ is the characteristic function, the locally integrable functions on the left hand side should be interpreted as distributions (see this page (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution)), and the limit should be taken in $\mathcal{D}^{\prime}(\mathbb{R})$ . It should also be noted that $\operatorname{p.\!v.}(\frac{1}{x})$ can be represented by a proper integral as

$\operatorname{p.\!v.}(\frac{1}{x})(u)=\int_{0}^{\infty}\frac{u(x)-u(-x)}{x},$

where we have used the fact that the integrand is continuous because of the differentiability at 0. In fact, this viewpoint can be used to somewhat vastly increase the set of functions for which this principal value is well-defined, such as functions that are integrable, satisfy a Lipschitz condition at 0, and whose behavior for large $x$ makes the integral converge at infinity.
2.

Let $\ln|t|$ be the distribution induced by the locally integrable function $\ln|t|:\mathbb{R}\to\mathbb{R}$ . Then, for the distributional derivative (http://planetmath.org/OperationsOnDistributions) $D$ , we have ([2], pp. 149)

$D(\ln|t|)=\operatorname{p.\!v.}(\frac{1}{x}).$

References

1 M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.
2 S. Igari, Real analysis - With an introduction to Wavelet Theory, American Mathematical Society, 1998.
3 J. Rauch, Partial Differential Equations, Springer-Verlag, 1991.

Title	Cauchy principal part integral
Canonical name	CauchyPrincipalPartIntegral
Date of creation	2013-03-22 13:46:04
Last modified on	2013-03-22 13:46:04
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	10
Author	mathcam (2727)
Entry type	Definition
Classification	msc 46F05
Classification	msc 46-00
Synonym	Cauchy principal value
Related topic	ImproperIntegral