Complex Analysis with Applications by Nakhlé H. Asmar & Loukas Grafakos

Author:Nakhlé H. Asmar & Loukas Grafakos
Language: eng
Format: epub
ISBN: 9783319940632
Publisher: Springer International Publishing

and using Cauchy’s generalized integral formula, we deduce

which completes the first part of the proof. The uniqueness of the power series is a consequence of Theorem 4.2.10.

Remark 4.3.3.

Since there is no guarantee that is the largest region on which the function f in Theorem 4.3.1 is defined, the series (4.3.2) may converge on an open disk larger than . Since a Taylor series is a power series, it follows from Theorem 4.2.5(iii) that the series (4.3.2) converges to f(z) absolutely on and uniformly on all closed subdisks of (in fact all closed subdisks of .) Also, in view of Corollary 4.2.9, the series may be differentiated term by term in as many times as we wish. This yields the Taylor series representation of the nth derivative

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