Cauchy principal part integral

Definition [1, 2, 3] Let C0() be the set of smooth functions with compact support on . Then the Cauchy principal part integral (or, more in line with the notation, the Cauchy principal value) p.v.(1x) is mapping p.v.(1x):C0() defined as


for uC0().

Theorem The mapping p.v.(1x) is a distribution of first order ( That is, p.v.(1x)𝒟1().

(proof. (

0.0.1 Properties

  1. 1.

    The distribution p.v.(1x) is obtained as the limit ([3], pp. 250)


    as n. Here, χ is the characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath, the locally integrable functions on the left hand side should be interpreted as distributions (see this page (, and the limit should be taken in 𝒟(). It should also be noted that p.v.(1x) can be represented by a proper integral as


    where we have used the fact that the integrand is continuousMathworldPlanetmathPlanetmath 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 conditionMathworldPlanetmath at 0, and whose behavior for large x makes the integral converge at infinityMathworldPlanetmathPlanetmath.

  2. 2.

    Let ln|t| be the distribution induced by the locally integrable function ln|t|:. Then, for the distributional derivativePlanetmathPlanetmath ( D, we have ([2], pp. 149)



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