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Cauchy’s Theorem, Second Part

  • Serge Lang
Chapter
  • 2.8k Downloads
Part of the Graduate Texts in Mathematics book series (GTM, volume 103)

Abstract

We wish to give a general global criterion when the integral of a holo-morphic 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 y when
$$\int\limits_r {f = 0}$$
for all holomorphic functions f, and also describe what the value of this integral may be if not 0.

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Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

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

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