Cauchy principal value

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

the finite number

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

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

$\int_a^b f(x)\,dx=\pm\infty$

for any a < b and

$\int_b^c f(x)\,dx=\mp\infty$

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

or

the finite number

$\lim_{a\rightarrow\infty}\int_{-a}^a f(x)\,dx$

where

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

and

$\int_0^\infty f(x)\,dx=\mp\infty$

(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

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

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:^[1]

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

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

1 Nomenclature
2 Examples
3 Distribution theory
4 References and notes
5 See also

[edit] 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..

[edit] Examples

Consider the difference in values of two limits:

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

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

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

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

Similarly, we have

$\lim_{a\rightarrow\infty}\int_{-a}^a\frac{2x\,dx}{x^2+1}=0,$

but

$\lim_{a\rightarrow\infty}\int_{-2a}^a\frac{2x\,dx}{x^2+1}=-\ln 4.$

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

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

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

[edit] 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

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

defined via the Cauchy principal value as

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