# Cauchy principal value

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Cauchy principal value

In mathematics, the Cauchy principal value of certain improper integrals, named after Augustin Louis Cauchy, is defined as either

* the finite number

::$lim_\left\{varepsilon ightarrow 0+\right\} left \left[int_a^\left\{b-varepsilon\right\} f\left(x\right),dx+int_\left\{b+varepsilon\right\}^c f\left(x\right),dx ight\right]$

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

::$int_a^b f\left(x\right),dx=pminfty$

:for any "a" < "b" and

::$int_b^c f\left(x\right),dx=mpinfty$

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

;or

* the finite number

::$lim_\left\{a ightarrowinfty\right\}int_\left\{-a\right\}^a f\left(x\right),dx$

:where

::$int_\left\{-infty\right\}^0 f\left(x\right),dx=pminfty$

:and

::$int_0^infty f\left(x\right),dx=mpinfty$

:(again, see plus or minus for precise usage of notation ±, ∓ ).

: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

::$lim_\left\{varepsilon ightarrow 0+\right\}int_\left\{b-frac\left\{1\right\}\left\{varepsilon^\left\{b-varepsilon\right\} f\left(x\right),dx+int_\left\{b+varepsilon\right\}^\left\{b+frac\left\{1\right\}\left\{varepsilonf\left(x\right),dx.$;or
* in terms of contour integrals of a complex-valued function "f (z)"; "z = x + i y", 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:cite book |author= Ram P. Kanwal |title=Linear Integral Equations: theory and technique |page= p. 191 |url =http://books.google.com/books?id=-bV9Qn8NpCYC&pg=PA194&lpg=PA194&dq=+%22Poincar%C3%A9-Bertrand+transformation%22&source=web&ots=iofB7oQccG&sig=2yieQ-eUpZTZtPcZrJJpBZAO-R4&hl=en#PPA191,M1
isbn=0817639403 |year=1996 |publisher=Birkhäuser |location=Boston |edition=2nd Edition
]

::$mathrm\left\{P\right\} int_\left\{L\right\} f\left(z\right) dz = int_L^* f\left(z\right) dz = lim_\left\{epsilon o 0 \right\} int_\left\{L\left( epsilon\right)\right\} f\left(z\right) dz ,$

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

Examples

Consider the difference in values of two limits:

:$lim_\left\{a ightarrow 0+\right\}left\left(int_\left\{-1\right\}^\left\{-a\right\}frac\left\{dx\right\}\left\{x\right\}+int_a^1frac\left\{dx\right\}\left\{x\right\} ight\right)=0,$

:$lim_\left\{a ightarrow 0+\right\}left\left(int_\left\{-1\right\}^\left\{-a\right\}frac\left\{dx\right\}\left\{x\right\}+int_\left\{2a\right\}^1frac\left\{dx\right\}\left\{x\right\} ight\right)=-ln 2.$

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

:$int_\left\{-1\right\}^1frac\left\{dx\right\}\left\{x\right\}\left\{ \right\}left\left(mbox\left\{which\right\} mbox\left\{gives\right\} -infty+infty ight\right).$

Similarly, we have

:$lim_\left\{a ightarrowinfty\right\}int_\left\{-a\right\}^afrac\left\{2x,dx\right\}\left\{x^2+1\right\}=0,$

but

:$lim_\left\{a ightarrowinfty\right\}int_\left\{-2a\right\}^afrac\left\{2x,dx\right\}\left\{x^2+1\right\}=-ln 4.$

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

:$int_\left\{-infty\right\}^inftyfrac\left\{2x,dx\right\}\left\{x^2+1\right\}\left\{ \right\}left\left(mbox\left\{which\right\} mbox\left\{gives\right\} -infty+infty ight\right).$

These pathologies do not afflict Lebesgue-integrable functions, that is, functions the integrals of whose absolute values are finite.

Distribution theory

Let $C_0^infty\left(mathbb\left\{R\right\}\right)$ be the set of smooth functions with compact support on the real line $mathbb\left\{R\right\}.$ Then, the map

: $operatorname\left\{p.!v.\right\}left\left(frac\left\{1\right\}\left\{x\right\} ight\right),: C_0^infty\left(mathbb\left\{R\right\}\right) o mathbb\left\{C\right\}$

defined via the Cauchy principal value as

:$operatorname\left\{p.!v.\right\}left\left(frac\left\{1\right\}\left\{x\right\} ight\right)\left(u\right)=lim_\left\{varepsilon o 0+\right\} int_\left\{| x|>varepsilon\right\} frac\left\{u\left(x\right)\right\}\left\{x\right\} , dx$ for $uin C_0^infty\left(mathbb\left\{R\right\}\right)$

is a distribution. This distribution appears for example in the Fourier transform of the Heaviside step function.

Nomenclature

The Cauchy principal value of a function $f$ can take on several nomenclatures, varying for different authors. These include (but are not limited to):

: $PV int f\left(x\right),dx,quad int_L^* f\left(z\right), dz,quad -!!!!!!int f\left(x\right),dx,$ $P ,$ P.V., $mathcal\left\{P\right\} ,$ $P_v ,$ $\left(CPV\right) ,$ and V.P.

*Augustin Louis Cauchy
*Sokhatsky-Weierstrass theorem

References and notes

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