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Encyclopedia > Cauchy principal value

In mathematics, the Cauchy principal value of certain improper integrals is defined as either Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ...

the finite number

$lim_{varepsilonrightarrow 0+} left[int_a^{b-varepsilon} f(x),dx+int_{b+varepsilon}^c f(x),dxright]$

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 "−").

the finite number

$lim_{arightarrowinfty}int_{-a}^a f(x),dx$

where

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

and

$int_0^infty f(x),dx=mpinfty$

(again, one sign is "+" and the other is "−").

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

$lim_{varepsilon rightarrow 0+}int_{b-frac{1}{varepsilon}}^{b-varepsilon} f(x),dx+int_{b+varepsilon}^{b+frac{1}{varepsilon}}f(x),dx.$

1 Nomenclature
2 Examples
3 Distribution theory
4 See also

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): $PV int f(x),dx$ , $P$ , P.V., $mathcal{P}$ , $P v$ , $(C P V)$ and V.P..

Examples

The Cauchy principal value of the otherwise ill-defined expression

$int_{-1}^1frac{dx}{x}{ } left(mbox{which} mbox{gives} -infty+inftyright).$

is given by considering the difference in values of two limits:

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

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

which leaves the principal value as $- ln2.$

Similarly, we have

$lim_{arightarrowinfty}int_{-a}^afrac{2x,dx}{x^2+1}=0,$

but

$lim_{arightarrowinfty}int_{-2a}^afrac{2x,dx}{x^2+1}=-ln 4.$

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

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

These pathologies do not afflict Lebesgue-integrable functions, that is, functions the integrals of whose absolute values are finite. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

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 In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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

defined via the Cauchy principal value as

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

is a distribution. This distribution appears for example in the Fourier transform of the Heaviside step function. In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

Cauchy principal value - Wikipedia, the free encyclopedia (184 words)
	In mathematics, the Cauchy principal value of certain improper integrals is defined as either
	The former is the Cauchy principal value of the otherwise ill-defined expression
	The former is the principal value of the otherwise ill-defined expression

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Contents

Examples

Distribution theory

See also