Editing Cauchy principal value

{{short description|Method for assigning values to certain improper integrals which would otherwise be undefined}}
{{about|a method for assigning values to improper integrals|the values of a complex function associated with a single branch|Principal value|the negative-power portion of a [[Laurent series]]|Principal part}}
In [[mathematics]], the '''Cauchy principal value''', named after [[Augustin-Louis Cauchy]], is a method for assigning values to certain [[improper integral]]s which would otherwise be undefined. In this method, a [[Singularity (mathematics)|singularity]] on an integral interval is avoided by limiting the integral interval to the non singular domain. 

==Formulation==
Depending on the type of [[Mathematical singularity|singularity]] in the integrand {{mvar|f}}, the Cauchy principal value is defined according to the following rules:

{{term|id=For a singularity at a finite number b|For a singularity at a finite number {{mvar|b}}}}{{defn|
<math display="block">\lim_{ \; \varepsilon \to 0^+ \;} \, \, \left[ \, \int_a^{b-\varepsilon} f(x) \, \mathrm{d}x ~ + ~ \int_{b+\varepsilon}^c  f(x) \, \mathrm{d}x \, \right]</math>
with <math> a < b < c </math> and where {{mvar|b}} is the difficult point, at which the behavior of the function {{mvar|f}} is such that
<math display="block">\int_a^b f(x)\,\mathrm{d}x = \pm\infty \quad</math> for any <math> a < b </math> and
<math display="block">\int_b^c f(x)\,\mathrm{d}x = \mp\infty \quad</math> for any <math> c > b </math>.
(See [[Plus–minus sign#Minus plus sign|''plus or minus'']] for the precise use of notations ± and ∓.)
}}
{{term|For a singularity at infinity (<math>\infty</math>)}}{{defn|
<math display="block">\lim_{a\to\infty} \, \int_{-a}^a f(x)\,\mathrm{d}x </math>
where <math display="block"> \int_{-\infty}^0 f(x) \,\mathrm{d}x = \pm\infty </math>
and <math display="block"> \int_0^\infty f(x) \,\mathrm{d}x = \mp\infty .</math>
}}

In some cases it is necessary to deal simultaneously with singularities both at a finite number {{mvar|b}} 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 [[Methods of contour integration|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 {{mvar|C}}. Define  <math>C(\varepsilon)</math> to be that same contour, where the portion inside the disk of radius {{mvar|ε}} around the pole has been removed. Provided the function <math>f(z)</math> is integrable over <math>C(\varepsilon)</math> no matter how small {{mvar|ε}} becomes, then the Cauchy principal value is the limit:<ref name=Kanwal>{{cite book |first=Ram P. |last=Kanwal |year=1996 |title=Linear Integral Equations: Theory and technique |edition=2nd |page=191 |publisher=Birkhäuser |place=Boston, MA |isbn=0-8176-3940-3 |url=https://books.google.com/books?id=-bV9Qn8NpCYC&q=+%22Poincar%C3%A9-Bertrand+transformation%22&pg=PA194 |via=Google Books}}</ref>
<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 integral|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 {{mvar|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 transform]]s.<ref name=King>{{cite book |first=Frederick W. |last=King |year=2009 |title=Hilbert Transforms |publisher=Cambridge University Press |place=Cambridge, UK |isbn=978-0-521-88762-5}}</ref>

== Distribution theory ==

Let <math> {C_{c}^{\infty}}(\mathbb{R}) </math> be the set of [[bump function]]s, i.e., the space of [[smooth function]]s with [[compact support]] on the [[real number|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 (mathematics)|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 [[distribution (mathematics)#Tempered distributions and Fourier transform|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]] [[integral kernel|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 transform]]s.

==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> <math>\mathcal{C},</math> and&nbsp;V.P.

== See also ==
*[[Hadamard finite part integral]]
*[[Hilbert transform]]
*[[Sokhotski–Plemelj theorem]]

== References ==
{{reflist|25em}}

[[Category:Augustin-Louis Cauchy]]
[[Category:Mathematical analysis]]
[[Category:Generalized functions]]
[[Category:Integrals]]
[[Category:Summability methods]]
[[Category:Schwartz distributions]]