# Cauchy principal value

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.

## Formulation

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined as one of the following:

1) The finite number
${\displaystyle \lim _{\varepsilon \rightarrow 0+}\left[\int _{a}^{b-\varepsilon }f(x)\,\mathrm {d} x+\int _{b+\varepsilon }^{c}f(x)\,\mathrm {d} x\right]}$
where b is a point at which the behavior of the function f is such that
${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=\pm \infty }$ for any a < b and
${\displaystyle \int _{b}^{c}f(x)\,\mathrm {d} x=\mp \infty }$ for any c > b
(see plus or minus for precise usage of notations ±, ∓).
2) The finite number
${\displaystyle \lim _{a\rightarrow \infty }\int _{-a}^{a}f(x)\,\mathrm {d} x}$
where ${\displaystyle \int _{-\infty }^{0}f(x)\,\mathrm {d} x=\pm \infty }$
and ${\displaystyle \int _{0}^{\infty }f(x)\,\mathrm {d} x=\mp \infty }$.
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
${\displaystyle \lim _{\varepsilon \rightarrow 0+}\left[\int _{b-{\frac {1}{\varepsilon }}}^{b-\varepsilon }f(x)\,\mathrm {d} x+\int _{b+\varepsilon }^{b+{\frac {1}{\varepsilon }}}f(x)\,\mathrm {d} x\right].}$
3) In terms of contour integrals

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:[1]

${\displaystyle \mathrm {P} \int _{L}f(z)\ \mathrm {d} z=\int _{L}^{*}f(z)\ \mathrm {d} z=\lim _{\varepsilon \to 0}\int _{L(\varepsilon )}f(z)\ \mathrm {d} z,}$
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, these definitions coincide with the standard definition of the integral.

Principal value integrals play a central role in the discussion of Hilbert transforms [2]

## Distribution theory

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

${\displaystyle \operatorname {p.\!v.} \left({\frac {1}{x}}\right)\,:\,{C_{c}^{\infty }}(\mathbb {R} )\to \mathbb {C} }$

defined via the Cauchy principal value as

${\displaystyle \left[\operatorname {p.\!v.} \left({\frac {1}{x}}\right)\right](u)=\lim _{\varepsilon \to 0^{+}}\int _{\mathbb {R} \setminus [-\varepsilon ;\varepsilon ]}{\frac {u(x)}{x}}\,\mathrm {d} x=\int _{0}^{+\infty }{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x\quad {\text{for }}u\in {C_{c}^{\infty }}(\mathbb {R} )}$

is a distribution. 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.

### Well-definedness as a distribution

To prove the existence of the limit

${\displaystyle \int _{0}^{+\infty }{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x}$
${\displaystyle \lim \limits _{x\searrow 0}u(x)-u(-x)=0}$ and hence
${\displaystyle \lim \limits _{x\searrow 0}{\frac {u(x)-u(-x)}{x}}=\lim \limits _{x\searrow 0}{\frac {u'(x)+u'(-x)}{1}}=2u'(0),}$

since ${\displaystyle u'(x)}$ is continuous and LHospitals rule applies.

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

${\displaystyle \left|\int \limits _{0}^{1}{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x\right|\leq \int \limits _{0}^{1}{\frac {|u(x)-u(-x)|}{x}}\,\mathrm {d} x\leq \int \limits _{0}^{1}{\frac {2x}{x}}\sup \limits _{x\in \mathbb {R} }|u'(x)|\,\mathrm {d} x\leq 2\sup \limits _{x\in \mathbb {R} }|u'(x)|}$.

As furthermore

${\displaystyle \left|\int \limits _{1}^{\infty }{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x\right|\leq 2\sup \limits _{x\in \mathbb {R} }|x\cdot u(x)|\int \limits _{1}^{\infty }{\frac {1}{x^{2}}}\,\mathrm {d} x=2\sup \limits _{x\in \mathbb {R} }|x\cdot u(x)|,}$

we note that the map ${\displaystyle \operatorname {p.\!v.} \left({\frac {1}{x}}\right)\,:\,{C_{c}^{\infty }}(\mathbb {R} )\to \mathbb {C} }$ is bounded by the usual seminorms for Schwartz functions ${\displaystyle u}$. Therefore this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.

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

### More general definitions

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

${\displaystyle xf=1\quad \Rightarrow \quad f=\operatorname {p.\!v.} \left({\frac {1}{x}}\right)+K\delta ,}$

where ${\displaystyle K}$ is a constant and ${\displaystyle \delta }$ 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 ${\displaystyle \mathbb {R} ^{n}}$. If ${\displaystyle K}$ 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

${\displaystyle [\operatorname {p.\!v.} (K)](f)=\lim _{\varepsilon \to 0}\int _{\mathbb {R} ^{n}\setminus B_{\varepsilon (0)}}f(x)K(x)\,\mathrm {d} x.}$

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 ${\displaystyle K}$ is a continuous homogeneous function of degree ${\displaystyle -n}$ whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.

## Examples

Consider the difference in values of two limits:

${\displaystyle \lim _{a\rightarrow 0+}\left(\int _{-1}^{-a}{\frac {\mathrm {d} x}{x}}+\int _{a}^{1}{\frac {\mathrm {d} x}{x}}\right)=0,}$
${\displaystyle \lim _{a\rightarrow 0+}\left(\int _{-1}^{-2a}{\frac {\mathrm {d} x}{x}}+\int _{a}^{1}{\frac {\mathrm {d} x}{x}}\right)=\ln 2.}$

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

${\displaystyle \int _{-1}^{1}{\frac {\mathrm {d} x}{x}}{\ }\left({\mbox{which}}\ {\mbox{gives}}\ -\infty +\infty \right).}$

Similarly, we have

${\displaystyle \lim _{a\rightarrow \infty }\int _{-a}^{a}{\frac {2x\,\mathrm {d} x}{x^{2}+1}}=0,}$

but

${\displaystyle \lim _{a\rightarrow \infty }\int _{-2a}^{a}{\frac {2x\,\mathrm {d} x}{x^{2}+1}}=-\ln 4.}$

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

${\displaystyle \int _{-\infty }^{\infty }{\frac {2x\,\mathrm {d} x}{x^{2}+1}}{\ }\left({\mbox{which}}\ {\mbox{gives}}\ -\infty +\infty \right).}$

## Nomenclature

The Cauchy principal value of a function ${\displaystyle f}$ can take on several nomenclatures, varying for different authors. Among these are:

${\displaystyle PV\int f(x)\,\mathrm {d} x,}$
${\displaystyle \int _{L}^{*}f(z)\,\mathrm {d} z,}$
${\displaystyle \int f(x)\,\mathrm {d} x,}$
as well as ${\displaystyle P,}$ P.V., ${\displaystyle {\mathcal {P}},}$ ${\displaystyle P_{v},}$ ${\displaystyle (CPV),}$ and V.P.