# Series of Analytic Functions and Singularities

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.