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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 Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
, is a method for assigning values to certain
improper integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
s 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 \lim_\, \lim_ \,\left ,\int_^ f(x)\,\mathrmx \,~ + ~ \int_^ f(x)\,\mathrmx \,\right In those cases where the integral may be split into two independent, finite limits, \lim_ \, \left, \,\int_a^ f(x)\,\mathrmx \,\\; < \;\infty and \lim_\;\left, \,\int_^c f(x)\,\mathrmx \,\ \; < \; \infty , 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 f(z) : z = x + i\, y \;, with x , y \in \mathbb \;, with a pole on a contour . Define C(\varepsilon) to be that same contour, where the portion inside the disk of radius around the pole has been removed. Provided the function f(z) is integrable over C(\varepsilon) no matter how small becomes, then the Cauchy principal value is the limit: \operatorname \int_ f(z) \,\mathrmz = \lim_ \int_ f(z)\, \mathrmz . In the case of Lebesgue-integrable 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), an ...
, these definitions coincide with the standard definition of the integral. If the function f(z) is ''
meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
'', the
Sokhotski–Plemelj theorem The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it (#Version for the real line, see below) is often used in physics, althoug ...
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 well ...
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 (\mathbb) 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 bump f ...
s, i.e., the space of
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
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 smallest ...
on the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), 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 ...
\mathbb . Then the map \operatorname \left( \frac \right) \,:\, (\mathbb) \to \mathbb defined via the Cauchy principal value as \left \operatorname \left( \frac \right) \rightu) = \lim_ \int_ \frac \, \mathrm x = \int_^ \frac \, \mathrm x \quad \text u \in (\mathbb) 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 avoi ...
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 argume ...
.


Well-definedness as a distribution

To prove the existence of the limit \int_^ \frac \, \mathrmx for a Schwartz function u(x), first observe that \frac is continuous on _\lim__\;_\Bigl[_u(x)_-_u(-x)_\Bigr.html" ;"title=", \infty), as \lim_ \; \Bigl[ u(x) - u(-x) \Bigr">, \infty), as \lim_ \; \Bigl[ u(x) - u(-x) \Bigr~= ~0 ~ and hence \lim_ \, \frac ~=~ \lim_ \, \frac ~=~ 2u'(0)~, since u'(x) is continuous and L'Hopital's rule applies. Therefore, \int_0^1 \, \frac \, \mathrmx exists and by applying the mean value theorem to u(x) - u(-x) , we get: : \left, \, \int_0^1\,\frac \,\mathrmx \,\ \;\leq\; \int_0^1 \frac \,\mathrmx \;\leq\; \int_0^1\,\frac\,\sup_\,\Bigl, u'(x)\Bigr, \,\mathrmx \;\leq\; 2\,\sup_\,\Bigl, u'(x)\Bigr, ~. And furthermore: : \left, \,\int_1^\infty \frac \,\mathrmx \,\ \;\leq\; 2 \,\sup_ \,\Bigl, x\cdot u(x)\Bigr, ~\cdot\;\int_1^\infty \frac \;=\; 2 \,\sup_\, \Bigl, x \cdot u(x)\Bigr, ~, we note that the map \operatorname\;\left( \frac \right) \,:\, (\mathbb) \to \mathbb is bounded by the usual seminorms for
Schwartz functions 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 ...
u. 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 Function (mathematics), functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. T ...
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 d ...
. Note that the proof needs u merely to be continuously differentiable in a neighbourhood of 0 and x\,u to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as u integrable with compact support and differentiable at 0.


More general definitions

The principal value is the inverse distribution of the function x and is almost the only distribution with this property: x f = 1 \quad \Leftrightarrow \quad \exists K: \; \; f = \operatorname \left( \frac \right) + K \delta, where K is a constant and \delta the Dirac distribution. In a broader sense, the principal value can be defined for a wide class of
singular integral In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, who ...
kernels Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
on the Euclidean space \mathbb^ . If K 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 operatorname (K)f) = \lim_ \int_ f(x) K(x) \, \mathrm x. 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 K 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 ''deg ...
of degree -n whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the
Riesz transform In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension ''d'' > 1. They are a type of singular integral operator, meaning that they a ...
s.


Examples

Consider the values of two limits: \lim_\left(\int_^\frac + \int_a^1\frac\right)=0, This is the Cauchy principal value of the otherwise ill-defined expression \int_^1\frac, \text +\infty \text. Also: \lim_\left(\int_^\frac+\int_^1\frac\right)=\ln 2. Similarly, we have \lim_\int_^a\frac=0, This is the principal value of the otherwise ill-defined expression \int_^\infty\frac \text +\infty \text. but \lim_\int_^a\frac=-\ln 4.


Notation

Different authors use different notations for the Cauchy principal value of a function f, among others: PV \int f(x)\,\mathrmx, \mathrm \int f(x)\,\mathrmx, \int_L^* f(z)\, \mathrmz, -\!\!\!\!\!\!\int f(x)\,\mathrmx, as well as P, P.V., \mathcal, P_v, (CPV), and V.P.


See also

*
Hadamard finite part integral In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by . showed that this ca ...
*
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 The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it (#Version for the real line, see below) is often used in physics, althoug ...


References

{{reflist, 25em Augustin-Louis Cauchy Mathematical analysis Generalized functions Integrals Summability methods