Cauchy principal value – Wikipedia

Method for assigning values to certain improper integrals which would otherwise be undefined

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.

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. This is usually done by a limit of the form

Let C c ( R ) {displaystyle {C_{c}^{infty }}(mathbb {R} )} be the set of bump functions, i.e., the space of smooth functions with compact support on the real line R {displaystyle mathbb {R} } . Then the map

To prove the existence of the limit

Therefore, 0 1 u ( x ) u ( x ) x d x {displaystyle int _{0}^{1},{frac {u(x)-u(-x)}{x}},mathrm {d} x} exists and by applying the mean value theorem to u ( x ) u ( x ) , {displaystyle u(x)-u(-x),} we get:

And furthermore:

we note that the map

Note that the proof needs u {displaystyle u} merely to be continuously differentiable in a neighbourhood of 0 and x u {displaystyle x,u} to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as u {displaystyle u} integrable with compact support and differentiable at 0.

The principal value is the inverse distribution of the function x {displaystyle x} and is almost the only distribution with this property:

In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space R n {displaystyle mathbb {R} ^{n}} . If K {displaystyle 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

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

Different authors use different notations for the Cauchy principal value of a function f {displaystyle f} , among others:

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Cauchy principal value - Wikipedia