Residue theorem

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 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:

Let $U$ be a simply connected open subset of the complex plane containing a finite list of points $a 1, ..., a n$ , and $f$ a function defined and holomorphic on $U \ {a1,...,an}$ . Let $γ$ be a closed rectifiable curve in $U$ which does not meet any of the $a k$ , and denote the winding number of $γ$ around $a k$ by $I(γ, a k)$ . The line integral of $f$ around $γ$ is equal to $2π i$ 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, $I(γ, a k) = 1$ if $a k$ is in the interior of $γ$ , and 0 if not, so

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