The Winding Number and the Residue Theorem

The homotopy form of Cauchy’s theorem enables one to calculate many integrals of the form % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca % WGMbGaamizaiaadQhaaSqaaiabeo7aNbqab0Gaey4kIipaaaa!3CB6! $$ \int\limits_\gamma {fdz} $$ where f is meromorphic and γ is a closed piecewise differentiable curve (it being assumed that the poles of. do not lie on Im (γ)). Formulae enabling one to do this include the so-called Cauchy formula (see §2, Theorem 2). It is, however, necessary to have some topological information about the location of the poles relative to γ. (To phrase it very vaguely, we must know how many times γ winds around a.) We begin with this topological material.