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 :

- with
*a*<*b*<*c*and where b is the difficult point, at which the behavior of the function f is such that

- with

- for any
*a*<*b*and

- for any

- for any
*c*>*b*.

- for any

- (See plus or minus for the precise use of notations ± and ∓ .)

- (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

In those cases where the integral may be split into two independent, finite limits,

- and

the end result is the same, but no longer matches the definition and technically is not called a "principal value".

The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function *f*(*z*) : *z* = *x* + *i y* , *x*, *y* ∈ ℝ , with a pole on a contour C . Define *C*(*ε*) to be that 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

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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

**^**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.**^**King, Frederick W. (2009).*Hilbert Transforms*. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-88762-5.