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.

Formulation

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:

For a singularity at the finite number b
with and where b is the difficult point, at which the behavior of the function f is such that
for any and
for any (See plus or minus for the precise use of notations ± and ∓.)
For a singularity at infinity ()
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

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:[1]
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.[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 Sign function and 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 L'Hopital's rule applies.

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

And 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 0 and to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions 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 values of two limits:

This is the Cauchy principal value of the otherwise ill-defined expression

Also:

Similarly, we have

This is the principal value of the otherwise ill-defined expression

but

Notation

Different authors use different notations for the Cauchy principal value of a function , among others:

as well as P.V., and V.P.

See also

References

  1. ^ Kanwal, Ram P. (1996). Linear Integral Equations: Theory and technique (2nd ed.). Boston, MA: Birkhäuser. p. 191. ISBN 0-8176-3940-3 – via Google Books.
  2. ^ King, Frederick W. (2009). Hilbert Transforms. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-88762-5.