## You are here

HomeCauchy principal part integral

## Primary tabs

# 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. That is, $\operatorname{p.\!v.}(\frac{1}{x})\in\mathcal{D}^{{\prime 1}}(\mathbb{R})$.

(proof.)

# 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), 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 $D$, 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.

## Mathematics Subject Classification

46F05*no label found*46-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella