Complex Variables

Francis J. Flanigan

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318486
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

A caution to mathematics professors: Complex Variables does not follow conventional outlines of course material. One reviewer noting its originality wrote: `A standard text is often preferred [to a superior text like this] because the professor knows the order of topics and the problems, and doesn't really have to pay attention to the text. He can go to class without preparation.` Not so here — Dr. Flanigan treats this most important field of contemporary mathematics in a most unusual way. While all the material for an advanced undergraduate or first-year graduate course is covered, discussion of complex algebra is delayed for 100 pages, until harmonic functions have been analyzed from a real variable viewpoint. Students who have forgotten or never dealt with this material will find it useful for the subsequent functions. In addition, analytic functions are defined in a way which simplifies the subsequent theory. Contents include: Calculus in the Plane, Harmonic Functions in the Plane, Complex Numbers and Complex Functions, Integrals of Analytic Functions, Analytic Functions and Power Series, Singular Points and Laurent Series, The Residue Theorem and the Argument Principle, and Analytic Functions as Conformal Mappings.
Those familiar with mathematics texts will note the fine illustrations throughout and large number of problems offered at the chapter ends. An answer section is provided. Students weary of plodding mathematical prose will find Professor Flanigan's style as refreshing and stimulating as his approach.

1

Calculus in the Plane

Section 1.1 DOMAINS IN THE xy-PLANE

1.1.0 Introduction

Here’s what we’ll do in the first few chapters:

1. We examine the geography of the xy-plane. Some of this will be familiar from basic calculus (for example, distance between points), some may be new to you (for example, the important notion of “domain”). We must also consider curves in the plane.

2. We consider real-valued functions u(x, y) defined in the plane. We will examine the derivatives (partial derivatives, gradient, directional derivatives) and integrals (line integrals, double integrals) of these functions. Most of (1) above will be necessary for (2). All this happens in this chapter.

3. We next focus attention on a particular kind of real-valued function u(x, y), the so-called harmonic function (Chapter 2). These are very interesting in their own right, have beautiful physical interpretations, and point the way to complex analytic functions.

4. At last (Chapter 3) we consider points (x, y) of the plane as complex numbers x + iy and we begin our study of complex-valued functions of a complex variable. This study occupies the rest of the book.

One disadvantage of this approach is the fact that complex numbers and complex analytic functions (our chief top...