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Concept
Residue theorem
Summary
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof.
The statement is as follows:
Let U be a simply connected open subset of the complex plane containing a finite list of points a1, ..., an,
U0 = U \ {a1, ..., an},
and a function f defined and holomorphic on U0. Let γ be a closed rectifiable curve in U0, and denote the winding number of γ around ak by I(γ, ak). The line integral of f around γ is equal to 2πi times the sum of residues of f at the points, each counted as many times as γ winds around the point:
If γ is a positively oriented simple closed curve, I(γ, ak) = 1 if ak is in the interior of γ, and 0 if not, therefore
with the sum over those ak inside γ.
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 whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of f dz along a Jordan curve γi with interior V. The requirement that f be holomorphic on U0 = U \ is equivalent to the statement that the exterior derivative d(f dz) = 0 on U0. Thus if two planar regions V and W of U enclose the same subset of , the regions V \ W and W \ V lie entirely in U0, and hence
is well-defined and equal to zero. Consequently, the contour integral of f dz along γj = ∂V is equal to the sum of a set of integrals along paths λj, each enclosing an arbitrarily small region around a single aj — the residues of f (up to the conventional factor 2πi) at . Summing over , we recover the final expression of the contour integral in terms of the winding numbers .
Official source
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