Editing Residue theorem (section)

==Statement of Cauchy's residue theorem==
{{See also|Residue (complex analysis)}}
The statement is as follows:

<blockquote>
[[File:Residue theorem illustration.svg|thumb|Illustration of the setting]]
'''Residue theorem''': Let <math>U</math> be a [[simply connected]] [[open subset]] of the [[complex plane]] containing a finite list of points <math>a_1, \ldots, a_n,</math> <math>U_0 = U \smallsetminus \{a_1, \ldots, a_n\},</math> and a function <math>f</math> [[holomorphic function|holomorphic]] on <math>U_0.</math> Letting <math>\gamma</math> be a closed [[rectifiable curve]] in <math>U_0,</math> and denoting the [[residue (complex analysis)|residue]] of <math>f</math> at each point <math>a_k</math> by <math>\operatorname{Res}(f, a_k)</math> and the [[winding number]] of <math>\gamma</math> around <math>a_k</math> by <math>\operatorname{I}(\gamma, a_k),</math> the line integral of <math>f</math> around <math>\gamma</math> is equal to <math>2\pi i</math> times the sum of residues, each counted as many times as <math>\gamma</math> winds around the respective point:

<math display=block>
\oint_\gamma f(z)\, dz = 2\pi i \sum_{k=1}^n \operatorname{I}(\gamma, a_k) \operatorname{Res}(f, a_k).
</math>

If <math>\gamma</math> is a [[Curve orientation|positively oriented]] [[Jordan curve|simple closed curve]], <math>\operatorname{I}(\gamma, a_k)</math> is <math>1</math> if <math>a_k</math> is in the interior of <math>\gamma</math> and <math>0</math> if not, therefore

<math display=block>
\oint_\gamma f(z)\, dz = 2\pi i \sum \operatorname{Res}(f, a_k)
</math>

with the sum over those <math>a_k</math> inside {{nobr|<math>\gamma.</math><ref>{{harvnb|Whittaker|Watson|1920|loc=§6.1|page=112}}.</ref>}}
</blockquote>

The relationship of the residue theorem to Stokes' theorem is given by the [[Jordan curve theorem]].  The general [[plane curve]] {{mvar|γ}} must first be reduced to a set of simple closed curves <math>\{\gamma_i\}</math> whose total is equivalent to <math>\gamma</math> for integration purposes; this reduces the problem to finding the integral of <math>f\, dz</math> along a Jordan curve <math>\gamma_i</math> with interior <math>V.</math> The requirement that <math>f</math> be holomorphic on <math>U_0 = U \smallsetminus \{a_k\}</math> is equivalent to the statement that the [[exterior derivative]] <math>d(f\, dz) = 0</math> on <math>U_0.</math>  Thus if two planar regions <math>V</math> and <math>W</math> of <math>U</math> enclose the same subset <math>\{a_j\}</math> of <math>\{a_k\},</math> the regions <math>V \smallsetminus W</math> and <math>W \smallsetminus V</math> lie entirely in <math>U_0,</math> hence

<math display=block>
\int_{V \smallsetminus W} d(f \, dz) - \int_{W \smallsetminus V} d(f \, dz)
</math>

is well-defined and equal to zero. Consequently, the contour integral of <math>f\, dz</math> along <math>\gamma_j = \partial V</math> is equal to the sum of a set of integrals along paths <math>\gamma_j,</math> each enclosing an arbitrarily small region around a single <math>a_j</math> — the residues of <math>f</math> (up to the conventional factor <math>2\pi i</math> at <math>\{a_j\}.</math> Summing over <math>\{\gamma_j\},</math> we recover the final expression of the contour integral in terms of the winding numbers <math>\{\operatorname{I}(\gamma, a_k)\}.</math>

In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.