Editing Cauchy principal value (section)

==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>