Winding Numbers and Cauchy’s Theorem

Abstract

We wish to give a general global criterion when the integral of a holomorphic function along a closed path is O. In practice, we meet two types of properties of paths: (1) properties of homotopy, and (2) properties having to do with integration, relating to the number of times a curve “winds” around a point, as we already saw when we evaluated the integral

$$ \int {\frac{1}{{\zeta- z}}} d\zeta$$

along a circle centered at z. These properties are of course related, but they also exist independently of each other, so we now consider those conditions on a closed path γ when

$$ \int_{\gamma } {f = 0}$$

for all holomorphic functions f, and also describe what the value of this integral may be if not 0.