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Series of Analytic Functions and Singularities

  • Nakhlé H. Asmar
  • Loukas Grafakos
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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