Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 5 / b

The residue theorem states that if a function is holomorphic on and inside a positively oriented simple closed contour except at finitely many isolated singularities $a_1,\dots ,a_N$ in its interior, thenNo singularity may lie on the contour. More generally, a closed contour has a winding-number-weighted sum of residues, provided it and its enclosed winding regions lie in the domain on which the isolated singularities are specified.