Cauchy principal value

In mathematics, the Cauchy principal value of certain improper integrals 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 "−").

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, 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

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

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

Consider the difference in values of two limits:

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

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

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

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

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.

Categories: Mathematical analysis

The contents of this article are licensed from Wikipedia.org under the GNU Free Documentation License. How to see transparent copy

Encyclopedia

Dictionary

Quotes

Cauchy principal value

Nomenclature

Examples

The Online Encyclopedia and Dictionary

Encyclopedia

Dictionary

Quotes

Cauchy principal value

Nomenclature

Examples