Residue (complex analysis)
Definition

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)}, Res a (f) {\displaystyle \operatorname {Res} _{a}(f)}, Res z = a f (z) {\displaystyle \mathop {\operatorname {Res} } _{z=a}f(z)} or res z = a f (z) {\displaystyle \mathop {\operatorname {res} } _{z=a}f(z)}, 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-a\vert <\delta }.

Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a of a Laurent series.

The concept can be used to provide contour integration values of certain contour integral problems considered in the residue theorem. According to the residue theorem, for a meromorphic function f {\displaystyle f}, the residue at point a k {\displaystyle a_{k}} is given as:

where γ {\displaystyle \gamma } is a positively oriented simple closed curve around a k {\displaystyle a_{k}} and not including any other singularities on or inside the curve.

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}.
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