Method for assigning values to certain improper integrals which would otherwise be undefined
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. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain.
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.