Complex Analysis pp 133-154 | Cite as

# Winding Numbers and Cauchy’s Theorem

Chapter

## Abstract

We wish to give a general global criterion when the integral of a holomorphic function along a closed path is 0. 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
along a circle centered at
for all holomorphic functions

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

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

*f*, and also describe what the value of this integral may be if not 0.## Keywords

Holomorphic Function Small Circle Power Series Expansion Closed Path Simple Closed Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1999