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| In [[complex analysis]], a field in mathematics, the '''residue theorem''', sometimes called '''Cauchy's residue theorem''' (one of [[List of things named after Augustin-Louis Cauchy|many things named after]] [[Augustin-Louis Cauchy]]), is a powerful tool to evaluate [[line integral]]s of [[analytic function]]s over closed curves; it can often be used to compute real integrals as well. It generalizes the [[Cauchy integral theorem]] and [[Cauchy's integral formula]]. From a geometrical perspective, it is a special case of the [[Stokes' theorem|generalized Stokes' theorem]].
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| [[Image:Residue theorem illustration.png|right|250px|thumb|Illustration of the setting.]]
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| The statement is as follows:
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| Suppose ''U'' is a [[simply connected]] [[open subset]] of the [[complex plane]], and ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> are finitely many points of ''U'' and ''f'' is a [[function (mathematics)|function]] which is defined and [[holomorphic function|holomorphic]] on ''U'' \ {''a''<sub>1</sub>,...,''a''<sub>''n''</sub>}. If γ is a [[rectifiable curve]] in ''U'' which does not meet any of the ''a''<sub>''k''</sub>, and whose start point equals its endpoint, then
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| :<math>\oint_\gamma f(z)\, dz =
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| 2\pi i \sum_{k=1}^n \operatorname{I}(\gamma, a_k)
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| \operatorname{Res}( f, a_k ). </math>
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| If γ is a [[Curve orientation|positively oriented]] [[Jordan curve|simple closed curve]], I(γ, ''a''<sub>''k''</sub>) = 1
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| if ''a''<sub>''k''</sub> is in the interior of γ, and 0 if not, so
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| :<math>\oint_\gamma f(z)\, dz =
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| 2\pi i \sum \operatorname{Res}( f, a_k ) </math>
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| with the sum over those ''k'' for which ''a''<sub>''k''</sub> is inside γ.
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| Here, Res(''f'', ''a''<sub>''k''</sub>) denotes the [[residue (complex analysis)|residue]] of ''f'' at ''a''<sub>''k''</sub>, and I(γ, ''a''<sub>''k''</sub>) is the [[winding number]] of the curve γ about the point ''a''<sub>''k''</sub>. This winding number is an [[integer]] which intuitively measures how many times the curve γ winds around the point ''a''<sub>''k''</sub>; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around ''a''<sub>''k''</sub> and 0 if γ doesn't move around ''a''<sub>''k''</sub> at all.
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| The relationship of the residue theorem to Stokes' theorem is given by the [[Jordan curve theorem]]. The general [[plane curve]] γ must first be reduced to a set of simple closed curves {γ<sub>''i''</sub>} whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of ''f'' ''dz'' along a Jordan curve γ<sub>''i''</sub> with interior ''V''. The requirement that ''f'' be holomorphic on ''U''<sub>0</sub> = ''U'' \ {''a''<sub>''k''</sub>} is equivalent to the statement that the [[exterior derivative]] ''d''(''f'' ''dz'') = 0 on ''U''<sub>0</sub>. Thus if two planar regions ''V'' and ''W'' of ''U'' enclose the same subset {''a''<sub>''j''</sub>} of {''a''<sub>''k''</sub>}, the regions ''V''\''W'' and ''W''\''V'' lie entirely in ''U''<sub>0</sub>, and hence <math>\scriptstyle\int_{V \backslash W} d(f \, dz) - \int_{W \backslash V} d(f \, dz)</math> is well-defined and equal to zero. Consequently, the contour integral of ''f'' ''dz'' along γ<sub>''j''</sub> = ∂V is equal to the sum of a set of integrals along paths λ<sub>''j''</sub>, each enclosing an arbitrarily small region around a single ''a''<sub>''j''</sub>—the residues of ''f'' (up to the conventional factor 2π''i'') at {''a''<sub>''j''</sub>}. Summing over {γ<sub>''j''</sub>}, we recover the final expression of the contour integral in terms of the winding numbers {I(γ, ''a''<sub>''k''</sub>)}.
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| 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.
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| ==Example==
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| The integral
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| :<math>\int_{-\infty}^\infty {e^{itx} \over x^2+1}\,dx</math>
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| [[Image:ContourDiagram_j.png|right|300px|thumb|The contour C.]]
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| arises in [[probability theory]] when calculating the [[characteristic function (probability theory)|characteristic function]] of the [[Cauchy distribution]]. It resists the techniques of elementary [[calculus]] but can be evaluated by expressing it as a limit of [[contour integral|contour integrals]].
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| Suppose ''t'' > 0 and define the contour ''C'' that goes along the [[real number|real]] line from −''a'' to ''a'' and then counterclockwise along a semicircle centered at 0 from ''a'' to −''a''. Take ''a'' to be greater than 1, so that the [[imaginary number|imaginary]] unit ''i'' is enclosed within the curve. The contour integral is
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| :<math>\int_C {f(z)}\,dz =\int_C {e^{itz} \over z^2+1}\,dz.</math>
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| Since ''e''<sup>''itz''</sup> is an [[entire function]] (having no [[mathematical singularity|singularities]] at any point in the complex plane), this function has singularities only where the denominator ''z''<sup>2</sup> + 1 is zero. Since ''z''<sup>2</sup> + 1 = (''z'' + ''i'')(''z'' − ''i''), that happens only where ''z'' = ''i'' or ''z'' = −''i''. Only one of those points is in the region bounded by this contour. Because ''f''(''z'') is
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| : <math>\begin{align}
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| \frac{e^{itz}}{z^2+1} & =\frac{e^{itz}}{2i}\left(\frac{1}{z-i}-\frac{1}{z+i}\right) \\
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| & =\frac{e^{itz}}{2i(z-i)} -\frac{e^{itz}}{2i(z+i)} ,
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| \end{align}</math>
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| the [[residue (complex analysis)|residue]] of ''f''(''z'') at ''z'' = ''i'' is
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| :<math>\operatorname{Res}\limits_{z=i}f(z)={e^{-t}\over 2i}.</math> | |
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| According to the residue theorem, then, we have
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| :<math>\int_C f(z)\,dz=2\pi i\cdot\operatorname{Res}\limits_{z=i}f(z)=2\pi i{e^{-t} \over 2i}=\pi e^{-t}.</math> | |
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| The contour ''C'' may be split into a "straight" part and a curved arc, so that
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| :<math>\int_{\mathrm{straight}} f(z)\,dz+\int_{\mathrm{arc}} f(z)\,dz=\pi e^{-t}\,</math> | |
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| and thus
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| :<math>\int_{-a}^a f(z)\,dz =\pi e^{-t}-\int_{\mathrm{arc}} f(z)\,dz.</math> | |
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| Using some [[Estimation lemma|estimations]], we have
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| :<math>\left|\int_{\mathrm{arc}}{e^{itz} \over z^2+1}\,dz\right| \leq \int_{\mathrm{arc}}\left|{e^{itz} \over z^2+1}\right| dz \le \int_{\mathrm{arc}}{1 \over |z^2+1|}dz\leq \int_{\mathrm{arc}}{1 \over a^2-1}dz = \frac{\pi a}{a^2-1}.</math>
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| and
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|
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| :<math>\lim_{a \to \infty} \frac{\pi a}{a^2-1} = 0.</math>
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| Note that, since ''t'' > 0 and for complex numbers in the upper halfplane the argument lies between 0 and π, one can estimate
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| :<math>\left|e^{itz}\right|=\left|e^{it|z|(\cos\phi + i\sin\phi)}\right|=\left|e^{-t|z|\sin\phi + it|z|\cos\phi}\right|=e^{-t|z|\sin\phi} \le 1.</math>
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| Therefore
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| :<math>\int_{-\infty}^\infty {e^{itz} \over z^2+1}\,dz=\pi e^{-t}.</math>
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| If ''t'' < 0 then a similar argument with an arc ''C' '' that winds around −''i'' rather than ''i'' shows that
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| [[Image:ContourDiagram2_j.png|right|300px|thumb|The contour C'.]]
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| :<math>\int_{-\infty}^\infty{e^{itz} \over z^2+1}\,dz=\pi e^t,</math>
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| and finally we have | |
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| :<math>\int_{-\infty}^\infty{e^{itz} \over z^2+1}\,dz=\pi e^{-\left|t\right|}.</math>
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| (If ''t'' = 0 then the integral yields immediately to elementary calculus methods and its value is π.)
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| ==See also==
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| * [[Cauchy's integral formula]]
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| * [[Jordan's lemma]]
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| * [[Methods of contour integration]]
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| * [[Morera's theorem]]
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| * [[Nachbin's theorem]]
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| * [[Residue at infinity]]
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| * [[Logarithmic form]]
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| ==References==
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| ; General references
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| {{refbegin}}
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| * {{citation|authorlink=Lars Ahlfors|first = Lars|last = Ahlfors|title = Complex Analysis|publisher = McGraw Hill|year = 1979|ISBN = 0-07-085008-9}}
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| * {{citation|first=Dragoslav|last=Mitronivić|first2=Jovan|last2=Kečkić|title=The Cauchy method of residues: Theory and applications|publisher=D. Reidel Publishing Company|year=1984|ISBN=90-277-1623-4}}
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| * {{citation|authorlink=Ernst Leonard Lindelöf|first = Ernst|last=Lindelöf|title=Le calcul des résidus et ses applications à la théorie des fonctions|publisher=Editions Jacques Gabay|year=1905|publication-date=1989|ISBN=2-87647-060-8}}
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| {{refend}}
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| ==External links==
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| * {{springer|title=Cauchy integral theorem|id=p/c020900}}
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| * [http://mathworld.wolfram.com/ResidueTheorem.html Residue theorem] in [[MathWorld]]
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| * [http://math.fullerton.edu/mathews/c2003/ResidueCalcMod.html Residue Theorem Module by John H. Mathews]
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| [[Category:Theorems in complex analysis]]
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| [[Category:Analytic functions|analytic functions]]
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