Complex Analysis with Applications pp 227-291 | Cite as

# 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.