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
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