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Concept
Cauchy principal value
Summary
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the singularity (so the singularity is not covered by the integral).
Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:
In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity. This is usually done by a limit of the form
In those cases where the integral may be split into two independent, finite limits,
and
then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value".
The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function with with a pole on a contour C. Define to be that same contour, where the portion inside the disk of radius ε around the pole has been removed. Provided the function is integrable over no matter how small ε becomes, then the Cauchy principal value is the limit:
In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral.
If the function is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over C with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals.
Principal value integrals play a central role in the discussion of Hilbert transforms.
Let be the set of bump functions, i.e., the space of smooth functions with compact support on the real line . Then the map
defined via the Cauchy principal value as
is a distribution.
Official source
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