Editing Residue (complex analysis) (section)

=== Application in residue theorem ===
{{Main|Residue theorem}}
For a [[meromorphic function]] <math>f</math>, with a finite set of singularities within a [[Curve orientation|positively oriented]] [[Jordan curve|simple closed curve]] <math>C</math> which does not pass through any singularity, the value of the contour integral is given according to [[residue theorem]], as:<math display="block">
\oint_C f(z)\, dz = 2\pi i \sum_{k=1}^n \operatorname{I}(C, a_k) \operatorname{Res}(f, a_k).
</math>where <math>\operatorname{I}(C, a_k)</math>, the winding number, is <math>1</math> if <math>a_k</math> is in the interior of <math>C</math> and <math>0</math> if not, simplifying to:<math display="block">
\oint_\gamma f(z)\, dz = 2\pi i \sum \operatorname{Res}(f, a_k)
</math>where <math>a_k</math> are all isolated singularities within the contour <math>C</math>.