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<vert z-avert <delta } . Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−1 of a Laurent series. The definition of a residue can be generalized to arbitrary Riemann surfaces. Suppose ω {displaystyle omega } is a 1-form on a Riemann surface. Let ω {displaystyle omega } be meromorphic at some point x {displaystyle x} , so that we may write ω {displaystyle omega } in local coordinates as f ( z ) d z {displaystyle f(z);dz} . Then the residue of ω {displaystyle omega } at x {displaystyle x} is defined to be the residue of f ( z ) {displaystyle f(z)} at the point corresponding to x {displaystyle x} . Computing the residue of a monomial makes most residue computations easy to do. Since path integral computations are homotopy invariant, we will let C {displaystyle C} be the circle with radius 1 {displaystyle 1} . Then, using the change of coordinates z → e i θ {displaystyle z o e^{i heta }} we find that