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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.

Formulation

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:

1) For a singularity at the finite number b:

limε→0+[∫ab−εf(x)dx+∫b+εcf(x)dx]displaystyle lim _varepsilon rightarrow 0^+left[int _a^b-varepsilon f(x),mathrm d x+int _b+varepsilon ^cf(x),mathrm d xright]

where b is a point at which the behavior of the function f is such that

∫abf(x)dx=±∞displaystyle int _a^bf(x),mathrm d x=pm infty for any a < b and

∫bcf(x)dx=∓∞displaystyle int _b^cf(x),mathrm d x=mp infty for any c > b

(see plus or minus for precise usage of notations ±, ∓).

2) For a singularity at infinity:

lima→∞∫−aaf(x)dxdisplaystyle lim _arightarrow infty int _-a^af(x),mathrm d x

where ∫−∞0f(x)dx=±∞displaystyle int _-infty ^0f(x),mathrm d x=pm infty

and ∫0∞f(x)dx=∓∞displaystyle 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

limε→0+[∫b−1εb−εf(x)dx+∫b+εb+1εf(x)dx].displaystyle lim _varepsilon rightarrow 0^+left[int _b-frac 1varepsilon ^b-varepsilon f(x),mathrm d x+int _b+varepsilon ^b+frac 1varepsilon f(x),mathrm d xright].

The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f(z); z = x + iy, with a pole on a contour C. Define C(ε) to be the same contour where the portion inside the disk of radius ε around the pole has been removed. Provided the function f(z) is integrable over C(ε) no matter how small ε becomes, then the Cauchy principal value is the limit:^[1]

P∫Cf(z) dz=limε→0+∫C(ε)f(z) dz,displaystyle mathrm P int _Cf(z) mathrm d z=lim _varepsilon to 0^+int _C(varepsilon )f(z) mathrm d z,

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 C 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 Cc∞(R)displaystyle 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 Rdisplaystyle mathbb R . Then the map

p.v.⁡(1x):Cc∞(R)→Cdisplaystyle operatorname p.!v. left(frac 1xright),:,C_c^infty (mathbb R )to mathbb C

defined via the Cauchy principal value as

[p.v.⁡(1x)](u)=limε→0+∫R∖[−ε;ε]u(x)xdx=∫0+∞u(x)−u(−x)xdxfor u∈Cc∞(R)displaystyle left[operatorname p.!v. left(frac 1xright)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 xquad textfor uin 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 Sign function and the Heaviside step function.

Well-definedness as a distribution

To prove the existence of the limit

∫0+∞u(x)−u(−x)xdxdisplaystyle int _0^+infty frac u(x)-u(-x)x,mathrm d x

for a Schwartz function u(x)displaystyle u(x), first observe that u(x)−u(−x)xdisplaystyle frac u(x)-u(-x)x is continuous on [0,∞)displaystyle [0,infty ), as

limx↘0u(x)−u(−x)=0displaystyle lim limits _xsearrow 0u(x)-u(-x)=0 and hence

limx↘0u(x)−u(−x)x=limx↘0u′(x)+u′(−x)1=2u′(0),displaystyle lim limits _xsearrow 0frac u(x)-u(-x)x=lim limits _xsearrow 0frac u'(x)+u'(-x)1=2u'(0),

since u′(x)displaystyle u'(x) is continuous and L'Hospital's rule applies.

Therefore, ∫01u(x)−u(−x)xdxdisplaystyle int limits _0^1frac u(x)-u(-x)x,mathrm d x exists and by applying the mean value theorem to u(x)−u(−x)displaystyle u(x)-u(-x), we get that

|∫01u(x)−u(−x)xdx|≤∫01|u(x)−u(−x)|xdx≤∫012xxsupx∈R|u′(x)|dx≤2supx∈R|u′(x)|int limits _0^1frac u(x)-u(-x)x,mathrm d xright.

As furthermore

|∫1∞u(x)−u(−x)xdx|≤2supx∈R|x⋅u(x)|∫1∞1x2dx=2supx∈R|x⋅u(x)|,int limits _1^infty frac u(x)-u(-x)x,mathrm d xright

we note that the map p.v.⁡(1x):Cc∞(R)→Cdisplaystyle operatorname p.!v. left(frac 1xright),:,C_c^infty (mathbb R )to mathbb C is bounded by the usual seminorms for Schwartz functions udisplaystyle 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 udisplaystyle u merely to be continuously differentiable in a neighbourhood of 0displaystyle 0 and xudisplaystyle xu to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as udisplaystyle u integrable with compact support and differentiable at 0.

More general definitions

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

xf=1⇒f=p.v.⁡(1x)+Kδ,displaystyle xf=1quad Rightarrow quad f=operatorname p.!v. left(frac 1xright)+Kdelta ,

where Kdisplaystyle K is a constant and δdisplaystyle 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 Rndisplaystyle mathbb R ^n. If Kdisplaystyle 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

[p.v.⁡(K)](f)=limε→0∫Rn∖Bε(0)f(x)K(x)dx.displaystyle [operatorname p.!v. (K)](f)=lim _varepsilon to 0int _mathbb R ^nsetminus 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 Kdisplaystyle K is a continuous homogeneous function of degree −ndisplaystyle -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:

lima→0+(∫−1−adxx+∫a1dxx)=0,displaystyle lim _arightarrow 0+left(int _-1^-afrac mathrm d xx+int _a^1frac mathrm d xxright)=0,

lima→0+(∫−1−2adxx+∫a1dxx)=ln⁡2.displaystyle lim _arightarrow 0+left(int _-1^-2afrac mathrm d xx+int _a^1frac mathrm d xxright)=ln 2.

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

∫−11dxx (which gives −∞+∞).displaystyle int _-1^1frac mathrm d xx left(mboxwhich mboxgives -infty +infty right).

Similarly, we have

lima→∞∫−aa2xdxx2+1=0,displaystyle lim _arightarrow infty int _-a^afrac 2x,mathrm d xx^2+1=0,

but

lima→∞∫−2aa2xdxx2+1=−ln⁡4.displaystyle lim _arightarrow infty int _-2a^afrac 2x,mathrm d xx^2+1=-ln 4.

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

∫−∞∞2xdxx2+1 (which gives −∞+∞).displaystyle int _-infty ^infty frac 2x,mathrm d xx^2+1 left(mboxwhich mboxgives -infty +infty right).

Notation

Different authors use different notations for the Cauchy principal value of a function fdisplaystyle f, among others:

PV∫f(x)dx,displaystyle PVint f(x),mathrm d x,

p.v.∫f(x)dx,displaystyle mathrm p.v. int f(x),mathrm d x,

∫L∗f(z)dz,displaystyle int _L^*f(z),mathrm d z,

−∫f(x)dx,displaystyle -!!!!!!int f(x),mathrm d x,

as well as P,displaystyle P, P.V., P,displaystyle mathcal P, Pv,displaystyle P_v, (CPV),displaystyle (CPV), and V.P.

See also

• Hadamard finite part integral


• Hilbert transform


• Sokhotski–Plemelj theorem

References