Analytic Functions and Their Properties

Abstract

This chapter is devoted to the study of analytic functions and their properties. In Sect. 2.1, we introduce the first definition that radically separates the behavior of real and complex functions—the definition of complex-differentiability. In spite of the formal similarity between the definitions of real- and complex-differentiability, the presented elementary examples show that complex-differentiability is a much stricter requirement on complex function than just differentiability of its real and imaginary parts. The concepts of holomorphy and analyticity of a function are also considered in this section, and the first step in the investigation of the fundamental question of the relationship between differentiability and analyticity is made: it is shown that a function analytic at a point is holomorphic at this point.