Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue of a function at a point of its domain is a complex number proportional 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 ⁠${\displaystyle f:\mathbb {C} \smallsetminus \{a_{k}\}_{k}\rightarrow \mathbb {C} }$⁠ that is holomorphic except at the discrete points ⁠${\displaystyle \{a_{k}\}_{k}}$⁠, which may include essential singularities.) Residues are typically readily computed and, once known, allow the determination of general contour integrals via the residue theorem.