C H A P T E R S E V E N
The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory
7.1 Introduction With the completion of the previous chapter we have really, I think, done pretty much everything we can do with just the imaginary Ï21 itself, and its extension to complex numbers. To continue on, the next logical step is to consider functions of variables that are complex valued, i.e., functions f (z) where z 5 x 1 iy. But then the question is, where do I stop? There is, today, an absolutely enormous literature in complex function theory, much of it purely mathematical, and just as much again of a practical, applications-oriented nature. Physicists and engineers, I believe, probably use complex variables and function theory more than do mathematicians. This book is not a textbook, but to jump into all of what we could jump into would turn this book into a huge textbook, perhaps one with a couple of thousand pages. You wouldn’t buy it, and I certainly don’t believe I would live long enough to write it. So what I am going to do is show you just the very beginnings of modern complex function theory, complex integration, as initiated by the French genius Augustin-Louis Cauchy (1789–1857) in an 1814 memoir to the French Acad´emie des Sciences.1 I am going to do this for three reasons. First, as I’ve already stated, we can’t do everything in a book like this one. Second, to begin at the beginning is really what I should do in a book that has a strong historical flavor. And third, I’m going to tell you about Cauchy’s initial work simply because it is so beautiful. When I first went off to Stanford to study electrical engineering, a story I told in the preface, I naturally began to take all the usual engineering math courses, such as differential and integral calculus, ordinary and partial differential equations, and even set theory. All of these courses were interesting to me, and in each one I quickly came to appreciate the power I was gaining in my ability to solve difficult problems. But it wasn’t until my first course in complex function theory, in the autumn of 1960, that I experienced a totally 187