# 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