Laurent Series: Isolated Singularities of Analytic Functions

Abstract

In this chapter, we continue the study of power series, but already their generalizations, namely power series containing terms $(z -z_0)^n$ with a negative integer n. These series were introduced by the French mathematician Pierre Laurent (1813–1854) in 1843. Laurent series are a valuable tool for studying the behavior of analytic functions near their isolated singularities, a classification of which is given here. It is noteworthy that, knowing the behavior of an analytic function near its singular points, one can determine its behavior in the entire domain, as well as calculate other characteristics associated with that function. As a result, it became possible to classify analytic functions according to their isolated singularities (Sect. 6.5). Interestingly, Laurent series have an equivalent relationship to Fourier series (Sect. 6.2), which have real applications in engineering (signal processing, spectroscopy, computer tomography, and many others).