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Series of Analytic Functions and Singularities

  • Nakhlé H. Asmar
  • Loukas GrafakosEmail author
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

In calculus, we use Taylor series to represent functions on intervals centered at fixed points with a radius of convergence that could be positive, infinite, or zero, depending on the remainder associated with the function. For example, \(\cos x\), \(e^x\), \(\frac{1}{1+x^2}\), and the function defined by \(e^{-1/x^2}\) for \(x\ne 0\) and 0 if \(x=0\) are all infinitely differentiable for all real x. The radius of convergence of the Taylor series representation around zero is \(\infty \) for the first two, 1 for the third one, and 0 for the last one. However in complex analysis, Taylor series are much nicer, in the sense that the remainder will play no role in determining their convergence. If a function is analytic on a disk of radius R centered at \(z_0\), then it has a Taylor series representation centered at \(z_0\) with radius at least R. For example the function \( \frac{1}{1+z^2}\) is analytic on the disk \(|z|<1\) but, as it is not differentiable at \(z=\pm i\), we do not expect the series to have a radius of convergence larger than 1.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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