Abstract
This chapter discusses the theory of functions of a complex variable and analytic functions, which extends the concepts of calculus to the complex plane. In doing so, differentiation and integration acquire new depth and elegance, and the two-dimensional nature of the complex plane yields many results useful in applied mathematics. Polynomials and rational functions in a real variable yield analytic functions when the real variable is replaced by z, where z = x + iy, and x and y are real numbers and i is the imaginary unit satisfying the property i2 = – 1. This is by no means an isolated example. In fact all elementary functions in calculus, such as exponentials, logarithms, and trigonometric functions, give rise to analytic functions when suitably extended to the complex plane. Therefore, the chapter presents the extensions of these elementary functions and their properties.
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