Cauchy principal part integral
Definition [1, 2, 3] Let ${C}_{0}^{\mathrm{\infty}}(\mathbb{R})$ 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) $\mathrm{p}.\mathrm{v}.(\frac{1}{x})$ is mapping $\mathrm{p}.\mathrm{v}.(\frac{1}{x}):{C}_{0}^{\mathrm{\infty}}(\mathbb{R})\to \u2102$ defined as
$$\mathrm{p}.\mathrm{v}.(\frac{1}{x})(u)=\underset{\epsilon \to 0+}{lim}{\int}_{x>\epsilon}\frac{u(x)}{x}\mathit{d}x$$ 
for $u\in {C}_{0}^{\mathrm{\infty}}(\mathbb{R})$.
Theorem The mapping $\mathrm{p}.\mathrm{v}.(\frac{1}{x})$ is a distribution of first order (http://planetmath.org/Distribution4). That is, $\mathrm{p}.\mathrm{v}.(\frac{1}{x})\in {\mathcal{D}}^{\prime 1}(\mathbb{R})$.
(proof. (http://planetmath.org/Operatornamepvfrac1xIsADistributionOfFirstOrder))
0.0.1 Properties

1.
The distribution $\mathrm{p}.\mathrm{v}.(\frac{1}{x})$ is obtained as the limit ([3], pp. 250)
$$\frac{{\chi}_{nx}}{x}\to \mathrm{p}.\mathrm{v}.(\frac{1}{x}).$$ as $n\to \mathrm{\infty}$. 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 $\mathrm{p}.\mathrm{v}.(\frac{1}{x})$ can be represented by a proper integral as
$$\mathrm{p}.\mathrm{v}.(\frac{1}{x})(u)={\int}_{0}^{\mathrm{\infty}}\frac{u(x)u(x)}{x},$$ 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 welldefined, 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 $\mathrm{ln}t$ be the distribution induced by the locally integrable function $\mathrm{ln}t:\mathbb{R}\to \mathbb{R}$. Then, for the distributional derivative^{} (http://planetmath.org/OperationsOnDistributions) $D$, we have ([2], pp. 149)
$$D(\mathrm{ln}t)=\mathrm{p}.\mathrm{v}.(\frac{1}{x}).$$
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^{}, SpringerVerlag, 1991.
Title  Cauchy principal part integral 

Canonical name  CauchyPrincipalPartIntegral 
Date of creation  20130322 13:46:04 
Last modified on  20130322 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 4600 
Synonym  Cauchy principal value 
Related topic  ImproperIntegral 