Editing Residue (complex analysis) (section)

== Definition ==
The residue of a [[meromorphic function]] <math>f</math> at an [[isolated singularity]] <math>a</math>, often denoted <math>\operatorname{Res}(f,a)</math>, <math>\operatorname{Res}_a(f)</math>, <math>\mathop{\operatorname{Res}}_{z=a}f(z)</math> or <math>\mathop{\operatorname{res}}_{z=a}f(z)</math>, is the unique value <math>R</math> such that <math>f(z)- R/(z-a)</math> has an [[Analytic function|analytic]] [[antiderivative (complex analysis)|antiderivative]] in a [[punctured disk]] <math>0<\vert z-a\vert<\delta</math>.  

Alternatively, residues can be calculated by finding [[Laurent series]] expansions, and one can define the residue as the coefficient ''a''<sub>−1</sub> of a Laurent series.  

The concept can be used to provide contour integration values of certain contour integral problems considered in the [[residue theorem]]. According to the [[residue theorem]], for a [[meromorphic function]] <math>f</math>, the residue at point <math>a_k</math> is given as: 

: <math>\operatorname{Res}(f,a_k) = {1 \over 2\pi i} \oint_\gamma f(z)\,dz \, .</math>

where <math>\gamma</math> is a [[Curve orientation|positively oriented]] [[Jordan curve|simple closed curve]] around <math>a_k</math> and not including any other singularities on or inside the curve.

The definition of a residue can be generalized to arbitrary [[Riemann surfaces]]. Suppose <math>\omega</math> is a [[One-form|1-form]] on a Riemann surface. Let <math>\omega</math> be meromorphic at some point <math>x</math>, so that we may write <math>\omega</math> in local coordinates as <math>f(z) \; dz</math>. Then, the residue of <math>\omega</math> at <math>x</math> is defined to be the residue of <math>f(z)</math> at the point corresponding to <math>x</math>.