Fundamental Theorems about Holomorphic Functions
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.
KeywordsPower Series Maximum Principle Holomorphic Function Entire Function Convergence Theorem
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