Cauchy principal value


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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