Residue theorem

The residue theorem in complex analysis is a powerful tool to evaluate line integrals of analytic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.

The statement is as follows. Suppose "U" is a simply connected open subset of the complex plane, and "a"₁,...,"a"_"n" are finitely many points of "U" and "f" is a function which is defined and holomorphic on "U" {"a"₁,...,"a"_"n"}. If γ is a rectifiable curve in "U" which bounds the "a"_"k", but does not meet any and whose start point equals its endpoint, then

:$oint\_gamma\; f(z),\; dz\; =2pi\; i\; sum\_\{k=1\}^n\; operatorname\{I\}(gamma,\; a\_k)\; operatorname\{Res\}(\; f,\; a\_k\; ).$

If γ is a Jordan curve, I(γ, "a"_"k") = 1 and so :$oint\_gamma\; f(z),\; dz\; =2pi\; i\; sum\_\{k=1\}^n\; operatorname\{Res\}(\; f,\; a\_k\; ).$

Here, Res("f", "a"_"k") denotes the residue of "f" at "a"_"k", and I(γ, "a"_"k") is the winding number of the curve γ about the point "a"_"k". This winding number is an integer which intuitively measures how often the curve γ winds around the point "a"_"k"; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around "a"_"k" and 0 if γ doesn't move around "a"_"k" at all.

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. 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.

Example

The integral

:$int\_\{-infty\}^infty\; \{e^\{itx\}\; over\; x^2+1\},dx$

arises in probability theory when calculating the characteristic function of the Cauchy distribution, and it resists the techniques of elementary calculus. We willevaluate it by expressing it as a limit of contour integralsalong the contour "C" that goes along the realline from −"a" to "a" and then counterclockwise alonga semicircle centered at 0 from "a" to −"a". Take"a" to be greater than 1, so that the imaginaryunit "i" is enclosed within the curve. The contour integral is

:$int\_C\; \{f(z)\},dz\; =int\_C\; \{e^\{itz\}\; over\; z^2+1\},dz.$

Since "e"^"itz" is an entire function(having no singularitiesat any point in the complex plane), this function hassingularities only where the denominator"z"² + 1 is zero. Since"z"² + 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 thiscontour. Because "f"("z") is

:

the residue of"f"("z") at "z" = "i" is

:$operatorname\{Res\}\_\{z=i\}f(z)=\{e^\{-t\}over\; 2i\}.$

According to the residue theorem, then, we have

:$int\_C\; f(z),dz=2pi\; icdotoperatorname\{Res\}\_\{z=i\}f(z)=2pi\; i\{e^\{-t\}\; over\; 2i\}=pi\; e^\{-t\}.$

The contour "C" may be split into a "straight"part and a curved arc, so that

:$int\_\{mbox\{straight+int\_\{mbox\{arc=pi\; e^\{-t\},$

and thus

:$int\_\{-a\}^a\; =pi\; e^\{-t\}-int\_\{mbox\{arc.$

It can be shown that if "t" > 0 then

:$int\_\{mbox\{arc\{e^\{itz\}\; over\; z^2+1\},dz\; ightarrow\; 0\; mbox\{as\}\; a\; ightarrowinfty.$

Therefore if "t" > 0 then

:$int\_\{-infty\}^infty\{e^\{itz\}\; over\; z^2+1\},dz=pi\; e^\{-t\}.$

A similar argument with an arc that winds around −"i"rather than "i" shows that if "t" < 0 then

:$int\_\{-infty\}^infty\{e^\{itz\}\; over\; z^2+1\},dz=pi\; e^t,$

and finally we have

:$int\_\{-infty\}^infty\{e^\{itz\}\; over\; z^2+1\},dz=pi\; e^\{-left|t\; ight.$

(If "t" = 0 then the integral yields immediately to elementary calculus methods and its value is π.)

ee also

* Methods of contour integration
* Morera's theorem
* Nachbin's theorem

References

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External links

* [http://mathworld.wolfram.com/ResidueTheorem.html Residue theorem] in MathWorld
* [http://math.fullerton.edu/mathews/c2003/ResidueCalcMod.html Residue Theorem Module by John H. Mathews]