Английская Википедия:Cauchy principal value
Шаблон:Short description Шаблон:About In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain.
Formulation
Depending on the type of singularity in the integrand Шаблон:Mvar, the Cauchy principal value is defined according to the following rules:
Шаблон:TermШаблон:Defn Шаблон:TermШаблон:Defn
In some cases it is necessary to deal simultaneously with singularities both at a finite number Шаблон:Mvar and at infinity. This is usually done by a limit of the form <math display="block">\lim_{\;\eta \to 0^+}\, \lim_{\;\varepsilon \to 0^+} \,\left[\,\int_{b - \frac{1}{\eta}}^{b - \varepsilon} f(x)\,\mathrm{d}x \,~ + ~ \int_{b+\varepsilon}^{b + \frac{1}{\eta}} f(x)\,\mathrm{d}x \,\right].</math>
In those cases where the integral may be split into two independent, finite limits, <math display="block">\lim_{\; \varepsilon\to 0^+\;} \, \left|\,\int_a^{b-\varepsilon} f(x)\,\mathrm{d}x \,\right|\; < \;\infty </math> and <math display="block"> \lim_{\;\eta\to 0^+}\;\left|\,\int_{b+\eta}^c f(x)\,\mathrm{d}x \,\right| \; < \; \infty ,</math>
then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value".
The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function <math> f(z) : z = x + i\, y \;,</math> with <math> x , y \in \mathbb{R} \;,</math> with a pole on a contour Шаблон:Mvar. Define <math>C(\varepsilon)</math> to be that same contour, where the portion inside the disk of radius Шаблон:Mvar around the pole has been removed. Provided the function <math>f(z)</math> is integrable over <math>C(\varepsilon)</math> no matter how small Шаблон:Mvar becomes, then the Cauchy principal value is the limit:[1] <math display="block">\operatorname{p.\!v.} \int_{C} f(z) \,\mathrm{d}z = \lim_{\varepsilon \to 0^+} \int_{C( \varepsilon)} f(z)\, \mathrm{d}z .</math>
In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral.
If the function <math>f(z)</math> is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over Шаблон:Mvar with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals.
Principal value integrals play a central role in the discussion of Hilbert transforms.[2]
Distribution theory
Let <math> {C_{c}^{\infty}}(\mathbb{R}) </math> be the set of bump functions, i.e., the space of smooth functions with compact support on the real line <math> \mathbb{R} </math>. Then the map <math display="block"> \operatorname{p.\!v.} \left( \frac{1}{x} \right) \,:\, {C_{c}^{\infty}}(\mathbb{R}) \to \mathbb{C} </math> defined via the Cauchy principal value as <math display="block"> \left[ \operatorname{p.\!v.} \left( \frac{1}{x} \right) \right](u) = \lim_{\varepsilon \to 0^{+}} \int_{\mathbb{R} \setminus [- \varepsilon,\varepsilon]} \frac{u(x)}{x} \, \mathrm{d} x = \lim_{\varepsilon \to 0^{+}} \int_{\varepsilon}^{+ \infty} \frac{u(x) - u(- x)}{x} \, \mathrm{d} x \quad \text{for } u \in {C_{c}^{\infty}}(\mathbb{R}) </math> is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the sign function and the Heaviside step function.
Well-definedness as a distribution
To prove the existence of the limit <math display="block"> \lim_{\varepsilon \to 0^{+}} \int_{\varepsilon}^{+ \infty} \frac{u(x) - u(- x)}{x} \, \mathrm{d}x </math> for a Schwartz function <math>u(x)</math>, first observe that <math>\frac{u(x) - u(-x)}{x}</math> is continuous on <math>[0, \infty),</math> as <math display="block"> \lim_{\,x \searrow 0\,} \; \Bigl[ u(x) - u(-x) \Bigr] ~= ~0 ~</math> and hence <math display="block"> \lim_{x\searrow 0} \, \frac{u(x) - u(-x)}{x} ~=~ \lim_{\,x\searrow 0\,} \, \frac{u'(x) + u'(-x)}{1} ~=~ 2u'(0)~, </math> since <math>u'(x)</math> is continuous and L'Hopital's rule applies.
Therefore, <math>\int_0^1 \, \frac{u(x) - u(-x)}{x} \, \mathrm{d}x</math> exists and by applying the mean value theorem to <math>u(x) - u(-x) ,</math> we get:
- <math> \left|\, \int_0^1\,\frac{u(x) - u(-x)}{x} \,\mathrm{d}x \,\right|
\;\leq\; \int_0^1 \frac{\bigl|u(x)-u(-x)\bigr|}{x} \,\mathrm{d}x \;\leq\; \int_0^1\,\frac{\,2x\,}{x}\,\sup_{x \in \mathbb{R} }\,\Bigl|u'(x)\Bigr| \,\mathrm{d}x \;\leq\; 2\,\sup_{x \in \mathbb{R} }\,\Bigl|u'(x)\Bigr| ~. </math>
And furthermore:
- <math> \left| \,\int_1^\infty \frac {\;u(x) - u(-x)\;}{x} \,\mathrm{d}x \,\right| \;\leq\; 2 \,\sup_{x\in\mathbb{R}} \,\Bigl|x\cdot u(x)\Bigr|~\cdot\;\int_1^\infty \frac{\mathrm{d}x}{\,x^2\,} \;=\; 2 \,\sup_{x\in\mathbb{R}}\, \Bigl|x \cdot u(x)\Bigr| ~, </math>
we note that the map <math display="block"> \operatorname{p.v.}\;\left( \frac{1}{\,x\,} \right) \,:\, {C_{c}^{\infty}}(\mathbb{R}) \to \mathbb{C} </math> is bounded by the usual seminorms for Schwartz functions <math> u</math>. Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.
Note that the proof needs <math>u</math> merely to be continuously differentiable in a neighbourhood of 0 and <math> x\,u </math> to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as <math>u</math> integrable with compact support and differentiable at 0.
More general definitions
The principal value is the inverse distribution of the function <math> x </math> and is almost the only distribution with this property: <math display="block"> x f = 1 \quad \Leftrightarrow \quad \exists K: \; \; f = \operatorname{p.\!v.} \left( \frac{1}{x} \right) + K \delta, </math> where <math> K </math> is a constant and <math> \delta </math> the Dirac distribution.
In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space <math> \mathbb{R}^{n} </math>. If <math> K </math> has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal-value distribution is defined on compactly supported smooth functions by <math display="block"> [\operatorname{p.\!v.} (K)](f) = \lim_{\varepsilon \to 0} \int_{\mathbb{R}^{n} \setminus B_{\varepsilon}(0)} f(x) K(x) \, \mathrm{d} x. </math> Such a limit may not be well defined, or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if <math> K </math> is a continuous homogeneous function of degree <math> -n </math> whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.
Examples
Consider the values of two limits: <math display="block">\lim_{a \to 0+}\left(\int_{-1}^{-a}\frac{\mathrm{d}x}{x} + \int_a^1\frac{\mathrm{d}x}{x}\right)=0,</math>
This is the Cauchy principal value of the otherwise ill-defined expression <math display="block">\int_{-1}^1\frac{\mathrm{d}x}{x}, \text{ (which gives } {-\infty}+\infty \text{)}.</math>
Also: <math display="block">\lim_{a \to 0+}\left(\int_{-1}^{-2 a}\frac{\mathrm{d}x}{x}+\int_{a}^1\frac{\mathrm{d}x}{x}\right)=\ln 2.</math>
Similarly, we have <math display="block">\lim_{a \to \infty}\int_{-a}^a\frac{2x\,\mathrm{d}x}{x^2+1}=0,</math>
This is the principal value of the otherwise ill-defined expression <math display="block">\int_{-\infty}^\infty\frac{2x\,\mathrm{d}x}{x^2+1} \text{ (which gives } {-\infty}+\infty \text{)}.</math> but <math display="block">\lim_{a\to\infty}\int_{-2a}^a\frac{2x\,\mathrm{d}x}{x^2+1}=-\ln 4.</math>
Notation
Different authors use different notations for the Cauchy principal value of a function <math>f</math>, among others: <math display="block">PV \int f(x)\,\mathrm{d}x,</math> <math display="block">\mathrm{p.v.} \int f(x)\,\mathrm{d}x,</math> <math display="block">\int_L^* f(z)\, \mathrm{d}z,</math> <math display="block"> -\!\!\!\!\!\!\int f(x)\,\mathrm{d}x,</math> as well as <math>P,</math> P.V., <math>\mathcal{P},</math> <math>P_v,</math> <math>(CPV),</math> and V.P.
See also
References
