Cauchy principal value

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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.

Formulation

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined as one of the following:

1) The finite number
where b is a point at which the behavior of the function f is such that
for any a < b and
for any c > b
(see plus or minus for precise usage of notations ±, ∓).
2) The finite number
where
and .
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
3) In terms of contour integrals

of a complex-valued function f(z); z = x + iy, with a pole on the contour. The pole is enclosed with a circle of radius ε and the portion of the path outside this circle is denoted L(ε). Provided the function f(z) is integrable over L(ε) no matter how small ε becomes, then the Cauchy principal value is the limit:[1]

where two of the common notations for the Cauchy principal value appear on the left of this equation.

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.

Principal value integrals play a central role in the discussion of Hilbert transforms [2]

Distribution theory

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. 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 , first observe that is continuous on , as

and hence

since is continuous and LHospitals rule applies.

Therefore exists and by applying the mean value theorem to , we get that

.

As furthermore

we note that the map is bounded by the usual seminorms for Schwartz functions . Therefore this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.

Note that the proof needs merely to be continuously differentiable in a neighbourhood of and to be bounded towards infinity. The principal value therefore is defined on even weaker assumptuions such as integrable with compact support and differentiable at 0.

More general definitions

The principal value is the inverse distribution of the function and is almost the only distribution with this property:

where is a constant and the Dirac distribution.

In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space . If has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal-value distribution is defined on compactly supported smooth functions by

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 is a continuous homogeneous function of degree whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.

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 can take on several nomenclatures, varying for different authors. Among these are:

as well as P.V., and V.P.

See also

References

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