Complex Analysis - Maths KU

Preface
7.2 Preface
Philosophy This text grew out of chapters 17-20 in Advanced Engineering
Mathematics, Second Edition (Jones and Bartlett Publishers), by Dennis
G. Zill and the late Michael R. Cullen. This present work represents an expansion
and revision of that original material and is intended for use in either
a one-semester or a one-quarter course. Its aim is to introduce the basic principles
and applications of complex analysis to undergraduates who have no
prior knowledge of this subject.
The motivation to adapt the material from Advanced Engineering Mathematics
into a stand-alone text sprang from our dissatisfaction with the succession
of textbooks that we have used over the years in our departmental
undergraduate course offering in complex analysis. It has been our experience
that books claiming to be accessible to undergraduates were often written at a
level that was too advanced for our audience. The “audience” for our juniorlevel
course consists of some majors in mathematics, some majors in physics,
but mostly majors from electrical engineering and computer science. At our
institution, a typical student majoring in science or engineering does not take
theory-oriented mathematics courses in methods of proof, linear algebra, abstract
algebra, advanced calculus, or introductory real analysis. Moreover,
the only prerequisite for our undergraduate course in complex variables is
the completion of the third semester of the calculus sequence. For the most
part, then, calculus is all that we assume by way of preparation for a student
to use this text, although some working knowledge of differential equations
would be helpful in the sections devoted to applications. We have kept the
theory in this introductory text to what we hope is a manageable level, concentrating
only on what we feel is necessary. Many concepts are conveyed
in an informal and conceptual style and driven by examples, rather than the
formal definition/theorem/proof. We think it would be fair to characterize
this text as a continuation of the study of calculus, but also the study of the
calculus of functions of a complex variable. Do not misinterpret the preceding
words; we have not abandoned theory in favor of “cookbook recipes”; proofs
of major results are presented and much of the standard terminology is used.
Indeed, there are many problems in the exercise sets in which a student is
asked to prove something. We freely admit that any student—not just majors
in mathematics—can gain some mathematical maturity and insight by
attempting a proof. But we know, too, that most students have no idea how
to start a proof. Thus, in some of our “proof” problems, either the reader
ix