The reader is familiar with elementary functions studied in real calculus. One can define similar functions involving a complex variable z = x + iy which reduce to the real-valued functions when we put y = 0 so that z = x (real). Complex functions have a wide range of applications. Most of the properties of real functions, such as continuity and differentiability, hold in the case of complex functions. Some of them have interesting properties, not apparent in the case of real functions.
In this chapter, we define the elementary functions of a complex variable and study their properties. In the following, z = x + iy and w = u + iv in Cartesian coordinates, and z = reiθ and w = Reiϕ in polars.