Residue (mathematics)

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In complex analysis, the residue of a function f holomorphic in an open set \Omega with possible exception of a point z_0\in\Omega where the function may admit an isolated singularity, is a particular number describing behaviour of f around z_0.

More precisely, if a function f is holomorphic in a neighbourhood of z_0 (but not necessarily at z_0 itself), with either a removable singularity or a pole at z_0, then it can be represented as a Laurent series around this point, that is

f(z) = \sum_{n=-N}^\infty c_n (z-z_0)^n

with some N\in \mathbb{N}\cup\{\infty\} and coefficients c_n\in \mathbb{C}.

The coefficient c_{-1} is the residue of f at z_0, denoted as \mathrm{Res}(f,z_0) or \underset{z=z_0}{\mathrm{Res}}f(z).

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.

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