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Categories: Mathematical analysis

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

Science Fair Project Encyclopedia Contents Page

09-23-2007 01:00:40
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