Cauchy principal part integral
Definition [1, 2, 3] Let 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) is mapping defined as
The distribution is obtained as the limit (, pp. 250)
as . Here, 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 . It should also be noted that can be represented by a proper integral as
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 makes the integral converge at infinity.
- 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|
|Date of creation||2013-03-22 13:46:04|
|Last modified on||2013-03-22 13:46:04|
|Last modified by||mathcam (2727)|
|Synonym||Cauchy principal value|