The residue theorem, sometimes called Cauchy's 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. From a geometrical perspective, it is a special case of the generalized Stokes' theorem.
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
If γ is a positively oriented Jordan curve, I(γ, a_{k}) = 1 and so
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 many times 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.
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_{i}} 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_{i} with interior V. The requirement that f be holomorphic on U_{0} = U \ {a_{k}} is equivalent to the statement that the exterior derivative d(f dz) = 0 on U_{0}. Thus if two planar regions V and W of U enclose the same subset {a_{j}} of {a_{k}}, the regions V\W and W\V lie entirely in U_{0}, and hence is welldefined and equal to zero. Consequently, the contour integral of f dz along γ_{i}=∂V is equal to the sum of a set of integrals along paths λ_{j}, each enclosing an arbitrarily small region around a single a_{j}—the residues of f (up to the conventional factor 2πi) at {a_{j}}. Summing over {γ_{i}}, we recover the final expression of the contour integral in terms of the winding numbers {I(γ, a_{k})}.
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 halfcircle in the upper or lower halfplane. The integral over this curve can then be computed using the residue theorem. Often, the halfcircle part of the integral will tend towards zero as the radius of the halfcircle grows, leaving only the realaxis part of the integral, the one we were originally interested in.
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