Fundamental Theorems about Holomorphic Functions

  • Reinhold Remmert
Part of the Graduate in Texts Mathematics book series (GTM, volume 122)


Having led to the Cauchy integral formula and the Cauchy-Taylor representation theorem, the theory of integration in the complex plane will temporarily pass off of center-stage. The power of the two mentioned results has already become clear but this chapter will offer further convincing examples of this power. First off, in section 1 we prove and discuss the Identity Theorem, which makes a statement about the “cohesion among the values taken on by a holomorphic function.” In the second section we illuminate the holomorphy concept from a variety of angles. In the third, the Cauchy estimates are discussed. As applications of them we get, among other things, Liouville’s theorem and, in section 4, the convergence theorems of Weierstrass. The Open Mapping Theorem and the Maximum Principle are proved in section 5.


Power Series Maximum Principle Holomorphic Function Entire Function Convergence Theorem 
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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Reinhold Remmert
    • 1
  1. 1.Mathematisches InstitutUniversität MünsterMünsterGermany

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