Abstract
In this chapter we show that the analytic functions are exactly the functions that can be expanded in a convergent power series about any point. Since power series can be treated very much as polynomials, this provides a powerful tool for dealing with analytic functions. In Sections 1 and 2 we review infinite series and series of functions. Sections 3 through 6 contain the basic material on power series. In Section 7 we use power series to show that the zeros of an analytic function are isolated. This leads to the uniqueness principle for analytic functions. Section 8 contains a formal definition of analytic continuation, which can be omitted at first reading.
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© 2001 Springer Science+Business Media New York
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Gamelin, T.W. (2001). Power Series. In: Complex Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21607-2_5
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DOI: https://doi.org/10.1007/978-0-387-21607-2_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95069-3
Online ISBN: 978-0-387-21607-2
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