Cauchy principal value

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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 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.