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.
Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:
In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity.
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
to be that same contour, where the portion inside the disk of radius ε around the pole has been removed.
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.
Principal value integrals play a central role in the discussion of Hilbert transforms.
be the set of bump functions, i.e., the space of smooth functions with compact support on the real line
defined via the Cauchy principal value as
The map itself may sometimes be called the principal value (hence the notation p.v.).
is continuous and L'Hopital's rule applies.
exists and by applying the mean value theorem to
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.
The principal value therefore is defined on even weaker assumptions such as
integrable with compact support and differentiable at 0.
The principal value is the inverse distribution of the function
In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space
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
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.
This is the Cauchy principal value of the otherwise ill-defined expression
This is the principal value of the otherwise ill-defined expression
Different authors use different notations for the Cauchy principal value of a function