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.
Contents
Formulation
Depending on the type of singularity in the integrand f, the Cauchy principal value is defined as one of the following:
of a complex-valued function f(z); z = x + iy, with a pole on a contour C. Define C(ε) to be the same contour where the portion inside the disk of radius ε around the pole has been removed. Provided the function f(z) is integrable over C(ε) 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 f(z) 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.
Distribution theory
Let
defined via the Cauchy principal value as
is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the Heaviside step function.
Well-definedness as a distribution
To prove the existence of the limit
for a Schwartz function
since
Therefore,
As furthermore
we note that the map
Note that the proof needs
More general definitions
The principal value is the inverse distribution of the function
where
In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space
Such a limit may not be well defined, or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if
Examples
Consider the difference in values of two limits:
The former is the Cauchy principal value of the otherwise ill-defined expression
Similarly, we have
but
The former is the principal value of the otherwise ill-defined expression
Nomenclature
The Cauchy principal value of a function