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
Previous Page Next Page