Preface

Basic complex variables is a very popular subject among mathematics faculty and

students. There is one big theorem (the Cauchy Integral Theorem) with a somewhat

diﬃcult proof, but this is followed by a host of easy to prove consequences that are

both surprising and powerful. Furthermore, for undergraduates, the subject is

refreshingly new and different from the analysis courses which precede it.

Undergraduate complex variables is our favorite course to teach, and we teach

it as often as we can. Over a period of years we developed notes for use in such a

course. The course is a one-semester undergraduate course on complex variables at

the University of Utah. It is designed for junior level students who have completed

three semesters of calculus and have also had some linear algebra and at least one

semester of foundations of analysis.

Over the past several years, these notes have been expanded and modified to

serve other audiences as well. We have used the expanded notes to teach a course

in applied complex variables for engineering students as well a course at the first

year graduate level for mathematics graduate students. The topics covered, the

emphasis given to topics, and the choice of exercises were different for each of these

audiences.

When taught as a one-semester junior level course for mathematics students,

the course is a transitional course between freshman and sophomore level calculus,

linear algebra, and differential equations and the much more sophisticated senior

level mathematics courses taught at Utah. The students are expected to understand

definitions and proofs, and the exercises assigned will include proofs as well as

computations. The course moves at a leisurely pace, and the material covered

includes only Chapters 1, 2, and 3 and selected sections from Chapters 4, 5, and 6.

A full year course could easily cover the entire text.

When we teach a one-semester undergraduate course for engineers using these

notes, we cover essentially the same material as in the course for mathematics

majors, but not all the proofs are done in detail, and there is more emphasis on

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