Series of Analytic Functions and Singularities

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.