Cauchy principal value explained

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, the Cauchy principal value is defined according to the following rules:

In some cases it is necessary to deal simultaneously with singularities both at a finite number and at infinity. This is usually done by a limit of the form$$\backslash lim\_\backslash ,\; \backslash lim\_\; \backslash ,\backslash left[\backslash ,\backslash int\_\{b\; -\; \backslash frac\{1\}\{\backslash eta\}\}^\{b\; -\; \backslash varepsilon\}\; f(x)\backslash ,\backslash mathrm\{d\}x\; \backslash ,~\; +\; ~\; \backslash int\_\{b+\backslash varepsilon\}^\{b\; +\; \backslash frac\{1\}\{\backslash eta\}\}\; f(x)\backslash ,\backslash mathrm\{d\}x\; \backslash ,\backslash right].$$

In those cases where the integral may be split into two independent, finite limits,and

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

f(z) : z = x + i y , with x , y \in R , with a pole on a contour . Define C(\varepsilon) to be that same contour, where the portion inside the disk of radius around the pole has been removed. Provided the function f(z) is integrable over C(\varepsilon) no matter how small becomes, then the Cauchy principal value is the limit:^[1] $$\backslash operatorname\; \backslash int\_\; f(z)\; \backslash ,\backslash mathrmz\; =\; \backslash lim\_\; \backslash int\_\; f(z)\backslash ,\; \backslash mathrmz\; .$$

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

f(z) is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over 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

{C ^{inf ty} _{c} }(\mathbb) be the set of bump functions, i.e., the space of smooth functions with compact support on the real line R . Then the map$$\backslash operatorname\; \backslash left(\backslash frac\; \backslash right)\; \backslash ,:\backslash ,\; (\backslash mathbb)\; \backslash to\; \backslash mathbb$$defined via the Cauchy principal value as$$\backslash left[\backslash operatorname\{p.\backslash !v.\}\; \backslash left(\backslash frac\{1\}\{x\}\; \backslash right)\; \backslash right](u)\; =\; \backslash lim\_\; \backslash int\_\; \backslash frac\; \backslash ,\; \backslash mathrm\; x\; =\; \backslash lim\_\; \backslash int\_^\; \backslash frac\; \backslash ,\; \backslash mathrm\; x\; \backslash quad\; \backslash text\; u\; \backslash in\; (\backslash mathbb)$$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

u(x) , first observe that (u(x) - u( - x))/(x) is continuous on [ 0 , inf ty), as$$\backslash lim\_\; \backslash ;\; \backslash Bigl[u(x)\; -\; u(-x)\; \backslash Bigr]\; ~=\; ~0\; ~$$and hence $$\backslash lim\_\; \backslash ,\; \backslash frac\; ~=~\; \backslash lim\_\; \backslash ,\; \backslash frac\; ~=~\; 2u\text{'}(0)~,$$since u'(x) is continuous and L'Hopital's rule applies.

Therefore,

\int ^{1} _{0} (u(x) - u( - x))/(x) d x exists and by applying the mean value theorem to u(x) - u( - x) , we get:\left| \int ^{1 (u(x) - u( - x))/(x)} _{0} d x \right| \leq \int ^{1} _{0} (l |u(x) - u( - x) r |)/(x) d x \leq \int ^{1 (2 x)/(x)} _{0} \sup _{x} l |u'(x) r | d x \leq 2 \sup _{x} l |u'(x) r | ~.

And furthermore:

\left| \int ^{inf ty} _{1} (u(x) - u( - x))/(x) d x \right| \leq 2 \sup _{x\inR} l |x ⋅ u(x) r |~ ⋅ \int ^{inf ty} _{1} (d x)/(x ^{2}) = 2 \sup _{x\inR} l |x ⋅ u(x) r | ~, 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:$$x\; f\; =\; 1\; \backslash quad\; \backslash Leftrightarrow\; \backslash quad\; \backslash exists\; K:\; \backslash ;\; \backslash ;\; f\; =\; \backslash operatorname\; \backslash left(\backslash frac\; \backslash right)\; +\; K\; \backslash delta,$$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

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$$[\backslash operatorname\{p.\backslash !v.\}\; (K)](f)\; =\; \backslash lim\_\; \backslash int\_\; f(x)\; K(x)\; \backslash ,\; \backslash mathrm\; x.$$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 values of two limits:$$\backslash lim\_\backslash left(\backslash int\_^\backslash frac\; +\; \backslash int\_a^1\backslash frac\backslash right)=0,$$

This is the Cauchy principal value of the otherwise ill-defined expression$$\backslash int\_^1\backslash frac,\; \backslash text\; +\backslash infty\; \backslash text.$$

Also:$$\backslash lim\_\backslash left(\backslash int\_^\backslash frac+\backslash int\_^1\backslash frac\backslash right)=\backslash ln\; 2.$$

Similarly, we have$$\backslash lim\_\backslash int\_^a\backslash frac=0,$$

This is the principal value of the otherwise ill-defined expression$$\backslash int\_^\backslash infty\backslash frac\; \backslash text\; +\backslash infty\; \backslash text.$$but$$\backslash lim\_\backslash int\_^a\backslash frac=-\backslash ln\; 4.$$

Notation

Different authors use different notations for the Cauchy principal value of a function

f , among others:$$PV\; \backslash int\; f(x)\backslash ,\backslash mathrmx,$$$$\backslash mathrm\; \backslash int\; f(x)\backslash ,\backslash mathrmx,$$$$\backslash int\_L^*\; f(z)\backslash ,\; \backslash mathrmz,$$$$-\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !\backslash int\; f(x)\backslash ,\backslash mathrmx,$$as well as P, P.V., l{P}, P _{v,} (CPV), and V.P.

See also

• Hadamard finite part integral
• Hilbert transform
• Sokhotski–Plemelj theorem

Notes and References

1. Book: Kanwal, Ram P. . 1996 . Linear Integral Equations: Theory and technique . 2nd . 191 . Birkhäuser . Boston, MA . 0-8176-3940-3 . Google Books.
2. Book: King, Frederick W. . 2009 . Hilbert Transforms . Cambridge University Press . Cambridge, UK . 978-0-521-88762-5.