Residue (complex analysis)

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In mathematics, more specifically complex analysis, the residue is a complex number equal to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f: \mathbb{C}-\{a_k\} \rightarrow \mathbb{C} that is holomorphic except at the discrete points {ak}, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

Contents

Definition

The residue of a meromorphic function f at an isolated singularity a, often denoted \operatorname{Res}(f,a) is the unique value R such that f(z) − R / (za) has an analytic antiderivative in a punctured disk 0<\vert z-a\vert<\delta. Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a − 1 of a Laurent series.

Example

As an example, consider the contour integral

where C is some simple closed curve about 0.

Let us evaluate this integral without using standard integral theorems that may be available to us. Now, the Taylor series for ez is well-known, and we substitute this series into the integrand. The integral then becomes

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