Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue 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 f : C ∖ { a k } k → C {displaystyle f:mathbb {C} setminus {a_{k}}_{k} ightarrow mathbb {C} } that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. In mathematics, more specifically complex analysis, the residue 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 f : C ∖ { a k } k → C {displaystyle f:mathbb {C} setminus {a_{k}}_{k} ightarrow mathbb {C} } that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. The residue of a meromorphic function f {displaystyle f} at an isolated singularity a {displaystyle a} , often denoted Res ⁡ ( f , a ) {displaystyle operatorname {Res} (f,a)} or Res a ⁡ ( f ) {displaystyle operatorname {Res} _{a}(f)} , is the unique value R {displaystyle R} such that f ( z ) − R / ( z − a ) {displaystyle f(z)-R/(z-a)} has an analytic antiderivative in a punctured disk 0 < | z − a | < δ {displaystyle 0