Analytic Functions

Abstract

In this chapter we present the fundamentals of the theory of analytic functions, and we carry the theory fairly far. First we define the integral of a complex function of a complex variable on a curve in the complex plane (the plane of the independent variable). We then establish some special cases of Cauchy’s integral theorem, which states that the integral of a differentiable complex-valued function of a complex variable is zero when taken over a closed curve. We infer Cauchy’s integral formula and give the Laurent series expansion of a function of a complex variable in the neighborhood of an isolated singular point. At the end of the chapter, we analyze the behavior of an analytic function near an essential singularity. We prove that by an appropriate choice of a sequence converging to this singularity, the function can be made to approach any limit we please.