Chapter 1: Complex Numbers

Abstract

Preliminary considerations Historically, complex numbers were introduced to solve algebraic equations. It was observed that the algebraic equation x 2 + 1 = 0 possesses no real solutions and the notation ± √ −1 was used to represent the roots. It was the year 1545 when the mathematicians Giro-lamo Cardano 1 , Ferro Tartaglia 2 and Rafel Bombeli 3 first used the square root of negative numbers to represent roots of equations. In 1797 a Norwegian Surveyor Wessel 4 reported to the Danish Academy of Science on the geometric interpretation of complex numbers of the form z = a + ib. Later in 1806 Robert Argand 5 also produced an essay on the geometric interpretation of complex numbers. The Swiss mathematician Leonard Euler 6 was the first to introduce the symbol i to represent the imaginary component of a number. In 1831 Carl Fredrich Gauss 7 formalized the work of Wessel and Argand and introduced numbers z = x + iy where i is a symbol used to represent a pure imaginary number having the property that i 2 = −1. Numbers of the form z = x + iy, where x and y are real numbers, were called complex numbers and many mathematicians have contributed to the development of the theory of complex numbers and functions associated with these numbers. In particular, much of the early work in the complex domain was done by the mathematicians Cauchy 8 , Weierstrass 9 and Riemann 10. For the last two hundred years many applications of complex numbers and complex functions have been developed in the areas of science and engineering. For example, the study areas of mappings, integration, differentiation, solutions of algebraic equations, solutions of ordinary differential equations, solutions of partial differential equations , summation of series, sequences, fractals and potential theory are just a few of the many application areas where one can find complex variable theory employed. You will find examples from many of these application areas presented throughout this textbook.