Analytic Functions

Abstract

In this chapter and onwards, we study properties of complex-valued functions of a complex variable. It turns out that every complex-valued function is determined by the corresponding vector function from $\mathbb R^2$ into $\mathbb R^2$. This fact enables us to obtain some properties of complex-valued functions from the first section. Fundamentally new is the notion of differentiability of complex-valued functions, which we introduce in Sect. 2.2, although it formally coincides with the standard definition (from calculus) of the differentiability of real functions of one real variable. Complex-valued differentiable functions, which we will call analytic functions, have many remarkable and unexpected properties that do not exist for real-valued differentiable functions. For example, a complex-valued differentiable function necessarily has derivatives of all orders, and many of its properties are determined by its values on arbitrary sets that have a limit point inside. These functions are of great importance both in various branches of mathematics and in many applications. The study of their properties is the main goal of complex analysis. In this section, we prove a criterion for the differentiability of complex-valued functions, which includes equivalence to the Cauchy–Riemann equations. They are a system of two partial differential equations that relate the real and imaginary parts of a complex-valued function. This leads to the concept of conjugate harmonic functions in Sect. 2.3. In addition, using some properties of conjugate harmonic functions, the hydrodynamic interpretation of analytic functions is given in Sect. 2.4. The chapter ends with Sect. 2.5, which introduces conformal map**s as analytic functions with a nonzero derivative. It turns out that a conformal function at a point $z_0$ preserves angles between curves at $z_0$ and equally stretches all curves starting at $z_0$. These two properties of a conformal function are characterized by the argument and the modulus of its derivative at $z_0,$ respectively.