Winding Numbers and Cauchy’s Theorem

  • Serge Lang
Part of the Graduate Texts in Mathematics book series (GTM, volume 103)


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


Holomorphic Function Small Circle Power Series Expansion Closed Path Simple Closed Curve 
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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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