ABSTRACT
This chapter provides a new kind of infinite series expansion which generalizes the idea of the power series expansion of a holomorphic function about a (nonsingular) point. It classifies the behavior of holomorphic functions near an isolated singular point. The chapter discusses holomorphic function on a punctured domain, classification of singularities, removable singularities, poles, and essential singularities. The Riemann removable singularities theorem and the Casorati—Weierstrass theorem are analyzed. To discuss convergence of Laurent series, one must first make a general agreement as to the meaning of the convergence of a “doubly infinite” series. The Cauchy integral for an isolated singularity, existence of Laurent expansions, holomorphic functions with isolated singularities, and classification of singularities in terms of Laurent series are also discussed.
