Cauchy principal value

# Cauchy principal value

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Cauchy principal value, named after Augustin Louis Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

, is a method for assigning values to certain improper integral
Improper integral
In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....

s which would otherwise be undefined.

## Formulation

Depending on the type of singularity
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

in the integrand f, the Cauchy principal value is defined as one of the following:
• the finite number

where b is a point at which the behavior of the function f is such that

for any a < b and

for any c > b (one sign is "+" and the other is "−"; see plus or minus for precise usage of notations ±, ∓).

or
• the finite number

where

.
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

or
• in terms of contour integrals
Methods of contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....

of a complex-valued function f(z); z = x + iy, with a pole on the contour. The pole is enclosed with a circle of radius ε and the portion of the path outside this circle is denoted L(ε). Provided the function f(z) is integrable over L(ε) no matter how small ε becomes, then the Cauchy principal value is the limit:

where two of the common notations for the Cauchy principal value appear on the left of this equation.

In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

, these definitions coincide with the standard definition of the integral.

Principal value integrals play a central role in the discussion of Hilbert transforms

## Examples

Consider the difference in values of two limits:

The former is the Cauchy principal value of the otherwise ill-defined expression

Similarly, we have

but

The former is the principal value of the otherwise ill-defined expression

## Distribution theory

Let be the set of smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

s with compact support on the real line
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

Then, the map

defined via the Cauchy principal value as

is a distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

. 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 Heaviside step function
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

.

The principal value is not exclusively defined on smooth functions ; it is enough that u be integrable, with compact support and differentiable at point 0.

It is the inverse distribution of function x and is almost the only distribution with this property :
where K is a constant and δ the Dirac distribution.

More generally, the principal value can be defined for a wide class of singular integral
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...

kernels on the Euclidean space Rn. If K(x) 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 K is a continuous homogeneous function
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

of degree −n whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.

## Nomenclature

The Cauchy principal value of a function can take on several nomenclatures, varying for different authors. Among these are:
P.V. and V.P.

• Augustin Louis Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...