Introduction

Complex Made Simple is intended as a text on complex analysis at the

beginning graduate level — students who have already taken a course on

this topic may nonetheless be interested in the results in the second half of

the book, beginning somewhere around Chapter 16, and experts in the field

may be amused by the proof of the Big Picard Theorem in Chapter 20.

The main prerequisite is a course typically called "Advanced Calculus"

or "Analysis", including topics such as uniform convergence, continuity and

compactness in Euclidean spaces. A hypothetical student who has never

heard of complex numbers should begin with Appendices 1 and 2 (students

who are familiar with basic manipulations with complex numbers on an in-

formal level can skip Appendix 2, although many such students should prob-

ably read Appendix 1). Definitions and results concerning metric spaces are

summarized in Appendix 4, with most proofs left as exercises. We decided

not to include a similar summary of elementary point-set topology: General

topological spaces occur in only a few sections, dealing with Riemann sur-

faces (students unfamiliar with general topology can skip those sections or

pretend that a topological space is just a metric space). The only abstract

algebra required is a rudimentary bit of group theory (normal subgroups

and homomorphisms), while the deepest fact from linear algebra used in the

text is that similar matrices have the same eigenvalues.

Of course the analysis here is really no simpler than that in any other text

on the topic at the same level (although we hope we have made it simple to

understand). A more accurate title might be Complex Explained in Excruci-

ating Detail: Since our main intent is pedagogical, we place great emphasis

on motivation, attempting to distinguish clearly between clever ideas and

routine calculations, to explain what various results "really mean", to show

IX