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

::$lim\_\{varepsilon\; ightarrow\; 0+\}\; left\; [int\_a^\{b-varepsilon\}\; f(x),dx+int\_\{b+varepsilon\}^c\; f(x),dx\; ight]$

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

::$int\_a^b\; f(x),dx=pminfty$

:for any "a" < "b" and

::$int\_b^c\; f(x),dx=mpinfty$

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

;or

* the finite number

::$lim\_\{a\; ightarrowinfty\}int\_\{-a\}^a\; f(x),dx$

:where

::$int\_\{-infty\}^0\; f(x),dx=pminfty$

:and

::$int\_0^infty\; f(x),dx=mpinfty$

:(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

::$lim\_\{varepsilon\; ightarrow\; 0+\}int\_\{b-frac\{1\}\{varepsilon^\{b-varepsilon\}\; f(x),dx+int\_\{b+varepsilon\}^\{b+frac\{1\}\{varepsilonf(x),dx.$;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]

::$mathrm\{P\}\; int\_\{L\}\; f(z)\; dz\; =\; int\_L^*\; f(z)\; dz\; =\; lim\_\{epsilon\; o\; 0\; \}\; int\_\{L(\; epsilon)\}\; f(z)\; dz\; ,$

: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:

:$lim\_\{a\; ightarrow\; 0+\}left(int\_\{-1\}^\{-a\}frac\{dx\}\{x\}+int\_a^1frac\{dx\}\{x\}\; ight)=0,$

:$lim\_\{a\; ightarrow\; 0+\}left(int\_\{-1\}^\{-a\}frac\{dx\}\{x\}+int\_\{2a\}^1frac\{dx\}\{x\}\; ight)=-ln\; 2.$

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

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

Similarly, we have

:$lim\_\{a\; ightarrowinfty\}int\_\{-a\}^afrac\{2x,dx\}\{x^2+1\}=0,$

but

:$lim\_\{a\; ightarrowinfty\}int\_\{-2a\}^afrac\{2x,dx\}\{x^2+1\}=-ln\; 4.$

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

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

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

Distribution theory

Let $C\_0^infty(mathbb\{R\})$ be the set of smooth functions with compact support on the real line $mathbb\{R\}.$ Then, the map

: $operatorname\{p.!v.\}left(frac\{1\}\{x\}\; ight),:\; C\_0^infty(mathbb\{R\})\; o\; mathbb\{C\}$

defined via the Cauchy principal value as

:$operatorname\{p.!v.\}left(frac\{1\}\{x\}\; ight)(u)=lim\_\{varepsilon\; o\; 0+\}\; int\_\{|\; x|>varepsilon\}\; frac\{u(x)\}\{x\}\; ,\; dx$ for $uin\; C\_0^infty(mathbb\{R\})$

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):

: $PV\; int\; f(x),dx,quad\; int\_L^*\; f(z),\; dz,quad\; -!!!!!!int\; f(x),dx,$ $P\; ,$ P.V., $mathcal\{P\}\; ,$ $P\_v\; ,$ $(CPV)\; ,$ and V.P.

See also

*Augustin Louis Cauchy
*Sokhatsky-Weierstrass theorem

References and notes

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