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In mathematics, the Cauchy principal value of certain improper integrals, named after Augustin Louis Cauchy, is defined as either
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \int_a^b f(x)\,dx=\pm\infty
- for any a < b and
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- Failed to parse (Missing texvc executable; please see math/README to configure.): \int_b^c f(x)\,dx=\mp\infty
- for any c > b (one sign is "+" and the other is "−").
or
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{a\rightarrow\infty}\int_{-a}^a f(x)\,dx
- where
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- Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^0 f(x)\,dx=\pm\infty
- and
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{\varepsilon \rightarrow 0+}\int_{b-\frac{1}{\varepsilon}}^{b-\varepsilon} f(x)\,dx+\int_{b+\varepsilon}^{b+\frac{1}{\varepsilon}}f(x)\,dx.
Nomenclature
The Cauchy principal value of a function Failed to parse (Missing texvc executable; please see math/README to configure.): f
can take on several nomenclatures, varying for different authors. These include (but are not limited to): Failed to parse (Missing texvc executable; please see math/README to configure.): PV \int f(x)\,dx
, Failed to parse (Missing texvc executable; please see math/README to configure.): P , P.V., Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{P} , Failed to parse (Missing texvc executable; please see math/README to configure.): P_v , Failed to parse (Missing texvc executable; please see math/README to configure.): (CPV)
and V.P..
Examples
Consider the difference in values of two limits:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_a^1\frac{dx}{x}\right)=0,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-1}^1\frac{dx}{x}{\ } \left(\mbox{which}\ \mbox{gives}\ -\infty+\infty\right).
Similarly, we have
- Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{a\rightarrow\infty}\int_{-a}^a\frac{2x\,dx}{x^2+1}=0,
but
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
Distribution theory
Let Failed to parse (Missing texvc executable; please see math/README to configure.): C_0^\infty(\mathbb{R})
be the set of smooth functions with compact support on the real line Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}.
Then, the map
- Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{p.\!v.}\left(\frac{1}{x}\right)\,: C_0^\infty(\mathbb{R}) \to \mathbb{C}
defined via the Cauchy principal value as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{p.\!v.}\left(\frac{1}{x}\right)(u)=\lim_{\varepsilon\to 0+} \int_{| x|>\varepsilon} \frac{u(x)}{x} \, dx
for Failed to parse (Missing texvc executable; please see math/README to configure.): u\in C_0^\infty(\mathbb{R})
is a distribution. This distribution appears for example in the Fourier transform of the Heaviside step function.
See also
This article incorporates material from Cauchy principal part integral on PlanetMath, which is licensed under the GFDL.
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