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Complex Power Series

In: Complex Analysis

Author

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  • Taras Mel’nyk

    (Taras Shevchenko National University of Kyiv, Faculty of Mathematics and Mechanics)

Abstract

The main goal of this chapter is to show that analytic functions can be represented as infinite power series. The key to proving this theorem is the Cauchy integral formula established in the previous section. Here we generalize this formula for derivatives and prove the surprising fact that derivatives of analytic functions can be calculated by integration. Conversely, we will establish that the sum of a complex power series is an analytic function in the open disk where this series converges. This fact is then used to prove that an analytic function is infinitely differentiable. In addition, the reader can familiarize himself with the proofs of such remarkable statements as Liouville’s theorem, Maurer’s theorem, and the equivalence of three approaches to the construction of the theory of analytic functions. In the last section, applications of power series representations lead us to a statement about the coincidence of analytic functions when they coincide on some sequence, and to a statement characterising the isolated zeros of an analytic function and their concentration.

Suggested Citation

  • Taras Mel’nyk, 2023. "Complex Power Series," Springer Books, in: Complex Analysis, chapter 5, pages 107-130, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-39615-1_5
    DOI: 10.1007/978-3-031-39615-1_5
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