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Cauchy principal value

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Title: Cauchy principal value  
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Cauchy principal value

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.

Contents

  • Formulation 1
  • Distribution theory 2
    • Well-definedness as a distribution 2.1
    • More general definitions 2.2
  • Examples 3
  • Nomenclature 4
  • See also 5
  • References 6

Formulation

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined as one of the following:

1) The finite number
\lim_{\varepsilon\rightarrow 0+} \left[\int_a^{b-\varepsilon} f(x)\,\mathrm{d}x+\int_{b+\varepsilon}^c f(x)\,\mathrm{d}x\right]
where b is a point at which the behavior of the function f is such that
\int_a^b f(x)\,\mathrm{d}x=\pm\infty for any a < b and
\int_b^c f(x)\,\mathrm{d}x=\mp\infty for any c > b
(see plus or minus for precise usage of notations ±, ∓).
2) The infinite number
\lim_{a\rightarrow\infty}\int_{-a}^a f(x)\,\mathrm{d}x
where \int_{-\infty}^0 f(x)\,\mathrm{d}x=\pm\infty
and \int_0^\infty f(x)\,\mathrm{d}x=\mp\infty.
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+} \left[\int_{b-\frac{1}{\varepsilon}}^{b-\varepsilon} f(x)\,\mathrm{d}x+\int_{b+\varepsilon}^{b+\frac{1}{\varepsilon}}f(x)\,\mathrm{d}x \right].
3) In terms of contour integrals

of a complex-valued function f(z); z = x + iy, 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) \ \mathrm{d}z = \int_L^* f(z)\ \mathrm{d}z = \lim_{\varepsilon \to 0 } \int_{L( \varepsilon)} f(z)\ \mathrm{d}z,
where two of the common notations for the Cauchy principal value appear on the left of this equation.

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.

Principal value integrals play a central role in the discussion of Hilbert transforms.[2]

Distribution theory

Let {C_{c}^{\infty}}(\mathbb{R}) be the set of bump functions, i.e., the space of smooth functions with compact support on the real line \mathbb{R} . Then the map

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

defined via the Cauchy principal value as

\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 = \int_{0}^{+ \infty} \frac{u(x) - u(- x)}{x} \, \mathrm{d} x \quad \text{for } u \in {C_{c}^{\infty}}(\mathbb{R})

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 Heaviside step function.

Well-definedness as a distribution

To prove the existence of the limit

\int_{0}^{+ \infty} \frac{u(x) - u(- x)}{x} \, \mathrm{d} x

for a Schwartz function u(x) , first observe that \frac{u(x) - u(-x)}{x} is continuous on [0, \infty) , as

\lim\limits_{x \searrow 0} u(x) - u(-x) = 0 and hence
\lim\limits_{x\searrow 0} \frac{u(x) - u(-x)}{x} = \lim\limits_{x\searrow 0} \frac{u'(x) + u'(-x)}{1} = 2u'(0),

since u'(x) is continuous and LHospitals rule applies.

Therefore \int\limits_0^1 \frac{u(x) - u(-x)}{x} \, \mathrm dx exists and by applying the mean value theorem to u(x) - u(-x) , we get that

\left| \int\limits_0^1 \frac{u(x) - u(-x)}{x} \,\mathrm dx \right| \leq \int\limits_0^1 \frac{|u(x)-u(-x)|}{x} \,\mathrm dx \leq \int\limits_0^1 \frac{2x}{x} \sup\limits_{x \in \mathbb R} |u'(x)| \,\mathrm dx \leq 2 \sup\limits_{x \in \mathbb R} |u'(x)| .

As furthermore

\left| \int\limits_1^\infty \frac {u(x) - u(-x)}{x} \,\mathrm dx \right| \leq 2 \sup\limits_{x\in\mathbb R} |x\cdot u(x)| \int\limits_1^\infty \frac 1{x^2} \,\mathrm dx = 2 \sup\limits_{x\in\mathbb R} |x\cdot u(x)|,

we note that the map \operatorname{p.\!v.} \left( \frac{1}{x} \right) \,:\, {C_{c}^{\infty}}(\mathbb{R}) \to \mathbb{C} is bounded by the usual seminorms for Schwartz functions u. 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 u merely to be continuously differentiable in a neighbourhood of 0 and xu to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as u integrable with compact support and differentiable at 0.

More general definitions

The principal value is the inverse distribution of the function x and is almost the only distribution with this property:

x f = 1 \quad \Rightarrow \quad f = \operatorname{p.\!v.} \left( \frac{1}{x} \right) + K \delta,

where K is a constant and \delta 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 \mathbb{R}^{n} . If K 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

[\operatorname{p.\!v.} (K)](f) = \lim_{\varepsilon \to 0} \int_{\mathbb{R}^{n} \setminus B_{\varepsilon(0)}} f(x) K(x) \, \mathrm{d} x.

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 K is a continuous homogeneous function of degree -n whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.

Examples

Consider the difference in values of two limits:

\lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{\mathrm{d}x}{x}+\int_a^1\frac{\mathrm{d}x}{x}\right)=0,
\lim_{a\rightarrow 0+}\left(\int_{-1}^{-2 a}\frac{\mathrm{d}x}{x}+\int_{a}^1\frac{\mathrm{d}x}{x}\right)=\ln 2.

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

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

Similarly, we have

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

but

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

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

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

Nomenclature

The Cauchy principal value of a function f can take on several nomenclatures, varying for different authors. Among these are:

PV \int f(x)\,\mathrm{d}x,
\int_L^* f(z)\, \mathrm{d}z,
-\!\!\!\!\!\!\int f(x)\,\mathrm{d}x,
as well as P, P.V., \mathcal{P}, P_v, (CPV), and V.P.

See also

References

  1. ^ Ram P. Kanwal (1996). Linear Integral Equations: theory and technique (2nd ed.). Boston: Birkhäuser. p. 191.  
  2. ^ Frederick W. King (2009). Hilbert Transforms. Cambridge: Cambridge University Press.  
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