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 a 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
(See plus or minus for the precise use of notations ± and ∓.)
For a singularity at infinity ()
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,
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:
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.
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 definitionsedit
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 integralkernels 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.
Consider the values of two limits:
This is the Cauchy principal value of the otherwise ill-defined expression
Similarly, we have
This is the principal value of the otherwise ill-defined expression
Different authors use different notations for the Cauchy principal value of a function , among others: