Residue (mathematics)

This article is about complex mathematical analysis. For material residue, see Residue (material).

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 a 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) then it can be represented as the 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 $\displaystyle \mathop {\mathrm {Res} } _{z=z_{0}}f(z).$

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.