# Cauchy principal value

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:

**1) For a singularity at the finite number***b*:

- where
*b*is a point at which the behavior of the function*f*is such that

- where

- for any
*a*<*b*and

- for any

- for any
*c*>*b* - (see plus or minus for precise usage of notations ±, ∓).

- for any

**2) 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

The Cauchy principal value can also be defined in terms of contour integrals 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:[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 *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.[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'Hospital's 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 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 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

## Notation

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

- as well as P.V., and V.P.

## References

- Ram P. Kanwal (1996).
*Linear Integral Equations: theory and technique*(2nd ed.). Boston: Birkhäuser. p. 191. ISBN 0-8176-3940-3. - Frederick W. King (2009).
*Hilbert Transforms*. Cambridge: Cambridge University Press. ISBN 978-0-521-88762-5.