Abstract
This chapter discuses the Taylor's theorem and the converse of Taylor's theorem, namely, that convergent power series are in fact analytic functions in their domain of convergence. The chapter also describes the isolated singularities. A function f(z) is analytic on a domain 0 < |z – z0 | <R, but not analytic or even necessarily defined at z0. Isolated singularities are classified in three categories: removable singularities, poles, and essential singularities. In addition, the chapter also describes a procedure for obtaining analytic continuations and interpretation of the Riemann surface of a function. A procedure for obtaining analytic continuations begins by expanding the given expression into a Taylor series.
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