Mathematics
5021
Fall, 2009
Instructor: Edward N. Wilson
Cupples I, Room 18
E-mail: enwilson@math.wustl.edu
Office Tel: 935-6729 (O.K. to leave messages, better to use e-mail)
Office Hours: MWF 2-3 and by appointment
Class Meeting Times
and Place: MWF 1-2, Earth and Planetary Sciences, Room 102
Textbook: Functions of One Complex Variable, by John B. Conway
Springer-Verlag New York Inc., 1973
Other Textbooks: There are literally hundreds of textbooks on complex variables,
the majority of them being technique oriented, non-rigorous, and intended for undergraduates who have not had a rigorous course in real analysis. Of the remaining
dozens of books, the following are books which the instructor likes and which
5021 students may wish to consult if they find
hard to understand:
1. Eli
Stein and Jerald Marsden have written textbooks
which, while primarily intended for undergraduates, are fairly rigorous and
include very nice discussions of many topics covered only lightly in
2. The book Real and Complex Analysis, by Walter Rudin, is an excellent one. As the title suggests, this book simultaneously treats real analysis and complex analysis, thereby emphasizing that complex analysis is a subset of real analysis.
3. The
“bible” for introductory complex analysis is the the
book Complex Analysis, by Lars Ahlfors. Almost certainly this book has been used as a
text for courses like Math 5021 more than any other book and perhaps more than
all other books combined. Virtually
every complex analyst owns this book and “likes” it in one way or another. But
the notations used by Ahlfors are often confusing and
his “explanations” often are “murky” and overly abbreviated. As a result, it may be best for students to
browse through the treatment of certain topics by Ahlfors
(and, thereafter, deciding whether or not to buy the book) ONLY after having
already studied these topics using a book like that of
4. The
book Complex Variables: The Geometric Viewpoint, by Steven Krantz of
Prerequisites: All students will be expected to have taken a rigorous undergraduate analysis course, one covering properties of the real numbers, metric spaces, multi-dimensional Euclidean spaces, proving the Heine-Borel Theorem at least for Euclidean spaces but preferably for metric spaces, then using it to develop one-variable and multi-variable calculus rigorously including a careful proof of the Implicit/Inverse Function Theorem, possibly the Picard Theorem on Uniqueness and Existence of ODE system solutions, and a reasonably careful treatment of line and surface integrals. It will be presumed that students have seen proofs of standard interchange of limits theorems (easy versions of the Fubini Theorem, differentiation and integration term-by-term theorems, differentiation under the integral theorem, etc.) It’s desirable but not essential that students have been exposed to the ideas in Lebesgue integration theory.
A one-semester basic topology course and a one-semester course in linear algebra
are also pre-requisites. However, unlike real analysis, if certain
topics in topology or linear algebra weren’t covered in a student’s
undergraduate courses, this will very likely not be a big problem. Similarly, while some prior exposure to the
ideas in group theory, ring theory, and differential geometry will be very
helpful with certain complex variable topics, this isn’t essential.
Homework: Over the course of the semester, there will be roughly a dozen homework assignments, roughly one per week. Assignments will be made approximately one week before they are due and the due date will be mentioned at the time the assignment is made. It is O.K. for students to discuss homework problems with one another and to ask the instructor for hints. However, every student should write up solutions in her/his own
words and style, NOT COPYING FROM ANOTHER STUDENT. If it happens that two or more papers are essentially identical, the grader will be encouraged to divide the total number of points awarded by the number of collaborators. When an assignment is due in class on a certain date, it will be O.K. for students to slip their solutions under the instructor’s office door later that day. But once the instructor has passed on the papers to the course grader, no more papers will be accepted. STUDENTS SHOULD NEVER PUT THEIR PAPERS IN THE INSTRUCTOR’S MAILBOX SINCE, IF THEY DO, THE ODDS ARE VERY HIGH THE PAPERS WON’T BE FOUND IN TIME TO PASS ON TO THE GRADER
Examinations: There will be two exams, a take-home exam roughly halfway through the semester and a two-hour in-class final examination on a date and time set by the University. The take-home will be similar to homework assignments but will be graded by the instructor rather than by the homework grader and may include some “challenge” problems harder than most homework problems. The final examination will be very different with an attempt made to use it as a “practice exam” for the May Qualifying Exam. Likely two questions will ask for an essay describing a big topic covered during the course: important ideas, proofs of one or two key results, examples, etc. The remaining questions will be problems similar to homework problems.
Grades: Homework will count for 40% of the course grade, the take-home exam for 15%, and the final exam for the remaining 45%.
Topics and Course Style: Over the course of the year, we will cover most of the material in the textbook plus some extra topics not treated in the text. However, we won’t systematically follow the topic order used in the text nor will we feel obliged to use the same notations as in the textbook. To the contrary, proofs different from those in the textbook will often be given by the instructor. The instructor will make an effort to occasionally break up the usual theoretical math course format of Definition, Theorem, Proof, Next Definition, Theorem, Proof,…with some brief comments on the history of the subject, possibilities for applications, overlaps with other branches of mathematics, areas of active research, etc.
Preliminary Outline
of Topics for the Fall Semester:
1. We’ll begin with a definition of complex number systems slightly more general than that in Chapter 1 of the textbook. We’ll cover all of the topics in Chapter 1
plus a bit more and go on to define complex differentiability and the material in Chapter 3. Mostly for fun, we’ll see how the fact that complex polynomials are trivially complex differentiable gives an easy proof of the Fundamental Theorem of Algebra.
2. Instead of moving directly into Chapter 4 of the text
(Chapter 2 is a review of undergraduate topics which students should carefully
read but which we won’t discuss in class), we’ll dwell on some things only
lightly treated by Conway: properties of
various types of Moebius transformations, ways to picture Moebius
transformations, and most importantly, the way in which conformal maps on the
complex plane correspond via stereographic projections to conformal maps on the
Riemann sphere.
3. Chapter 4 in the text: the original Cauchy’s Theorem, the astonishing improvement on it by Goursat, some easy Corollaries, etc.
4. Although Cauchy’s Theorem is the foundation block for most of the beautiful theorems in complex analysis (including those covered in Chapters 5-6), we won’t go immediately into a discussion of these theorems, but will instead take time out to discuss harmonic functions, including the topics in Chapter 10. Historically, the concept of a harmonic function motivated the development of complex analysis and a great deal of the usefulness of complex analysis to mathematical physics and differential geometry stems from applications and tools for dealing with harmonic functions.
5. Do the material in Chapters 5-6. This material, along with that in Chapter 4, constitutes the core of complex analysis and everyone should know the theorems and be able to prove them.
6. Go through the proof of the Riemann Mapping Theorem following the treatment in Chapter 7 of the text. This Theorem is one of the most amazing results in all of mathematics but the proof is very long and difficult. As time permits, we may illustrate it with a fast treatment of Scharz-Christoffel transformations, a topic not covered in the text.
Spring Semester: We’ll do a
variety of specialized topics including most of the material in Chapters 8, 9,
11, 12 of the text plus some other topics not yet decided upon. Actually, the above topic list 1-6 is quite
ambitious for one semester. It could be
that we’ll have to defer some or all of topic 6 to the spring semester.