In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 , 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 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
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 . Define
to be that same contour, where the portion inside the disk of radius around the pole has been removed. Provided the function
is integrable over
no matter how small becomes, then the Cauchy principal value is the limit:
In the case of
Lebesgue-integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
functions, that is, functions which are integrable in
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, 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 with the mean-value of the integrals with the contour displaced slightly above and below, so that the
residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as wel ...
can be applied to those integrals.
Principal value integrals play a central role in the discussion of
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
s.
Distribution theory
Let
be the set of
bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bum ...
s, i.e., the space of
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. Then the map
defined via the Cauchy principal value as
is a
distribution Distribution may refer to:
Mathematics
* Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a vari ...
. 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
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avo ...
and the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
.
Well-definedness as a distribution
To prove the existence of the limit
for a
Schwartz function
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables o ...
, first observe that
is continuous on
as
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
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
and therefore a
tempered distribution
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives ...
.
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
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
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.
See also
*
Hadamard finite part integral
*
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
*
Sokhotski–Plemelj theorem
References
{{reflist, 25em
Augustin-Louis Cauchy
Mathematical analysis
Generalized functions
Integrals
Summability methods