chapter 6, Complex Variables

Excerpt

Historical Comments: The study of analytic functions of a complex variable is synonymous with the name of Augustin-Louis Cauchy (1789–1857), one of the great mathematicians. Cauchy entered the Ecole Polytechnic to study engineering, but, because of poor health, was advised by Lagrange and Laplace to study mathematics instead. One of the most prolific mathematicians of all time, he published more than 700 papers. Among the subjects on which he worked were determinants, ordinary differential equations, partial differential equations, and complex variable theory. Cauchy provided the first systematic study of the theory of limits and the first rigorous proof of the existence of solutions to first-order differential equations. He also developed the concept of convergence of an infinite series and the theory of functions of a complex variable. In fact, most of the important theorems of the complex integral are associated with his name.

Other prominent names associated with complex variables are Georg Friedrich Bernhard Riemann (1826–1866) and Karl Weierstrass (1815–1897). Riemann entered Göttingen to study theology, but instead received his Ph.D. in mathematics under Gauss. He became professor of mathematics there in 1859 but died in this post when he was only 39 years old. Riemann introduced the concept of the integral, as it is used in basic calculus courses, in connection with his work on Fourier series. He also developed the mathematical basis for Einstein's theory of relativity known now as Riemannian geometry. After studying law for four years at Bonn, Weierstrass likewise turned to mathematics. Weierstrass developed complex variable theory based upon the power series representation of functions, but his “reverse” approach to the theory failed to attract many followers. Today we approach complex variables by first laying the groundwork of the complex differential and integral calculus, after which power series and Laurent series are developed.

Our objective in this chapter is to provide a fairly detailed treatment of the differential calculus of complex variables up through the Cauchy-Riemann equations. In Chapter 7 we extend the analysis to include the complex integral and Laurent series. Because complex variables are used so extensively in engineering and physics applications, our discussion here and in Chapter 7 is generally more detailed than it is in other subjects. The analysis of elementary complex functions provides a natural means of mapping certain two-dimensional regions into other two-dimensional regions—a powerful concept that is further explored in Chapter 7 in connection with steady-state heat conduction and fluid flow.