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From Citizendium, the Citizens' Compendium

In complex analysis, the residue of a function f holomorphic in an open set Ω with possible exception of a point where the function may admit an isolated singularity, is a particular number describing behaviour of f around z₀.

More precisely, if a function f is holomorphic in a neighbourhood of z₀ (but not necessarily at z₀ itself), with either a removable singularity or a pole at z₀, then it can be represented as a Laurent series around this point, that is

with some and coefficients

The coefficient c _{− 1} is the residue of f at z₀, denoted as Res(f,z₀) or

Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex functions. For example, the residue allows to evaluate path integrals of the function f via the residue theorem. This technique finds many applications in real analysis as well.