linguistic:paraphrasing, structural:expansion, quality:corruption, linguistic:translation
Application in residue theorem

For a meromorphic function f {\displaystyle f}, with a finite set of singularities within a positively oriented simple closed curve C {\displaystyle C} which does not pass through any singularity, the value of the contour integral is given according to residue theorem, as: ∮ C f (z) d z = 2 π i ∑ k = 1 n I (C, a k) Res (f, a k). {\displaystyle \oint _{C}f(z)\,dz=2\pi i\sum _{k=1}^{n}\operatorname {I} (C,a_{k})\operatorname {Res} (f,a_{k}).} where I (C, a k) {\displaystyle \operatorname {I} (C,a_{k})}, the winding number, is 1 {\displaystyle 1} if a k {\displaystyle a_{k}} is in the interior of C {\displaystyle C} and 0 {\displaystyle 0} if not, simplifying to: ∮ γ f (z) d z = 2 π i ∑ Res (f, a k) {\displaystyle \oint _{\gamma }f(z)\,dz=2\pi i\sum \operatorname {Res} (f,a_{k})} where a k {\displaystyle a_{k}} are all isolated singularities within the contour C {\displaystyle C}.