Math 542 (Complex Analysis) Section D1 Fall 2007
Course Description
This is the first graduate course in complex analysis, one of the most
elegant and beautiful branches of modern mathematics. Much of the
power of the subject stems from the close interplay between
algebra, analysis and geometry which greatly informs its development.
Topics which we will discuss include: the Cauchy--Riemann equations,
analytic and meromorphic functions, conformal mappings, the local and
global Cauchy theory, Liouville's theorem, the maximum modulus theorem
and the Schwarz Lemma, the classification of isolated singularities,
residue theory with applications, normal families, Montel's theorem,
the Riemann mapping theorem, infinite products and entire functions,
the Weierstrass factorization theorem, analytic continuation, Runge's
approximation theorem, the Dirichlet problem for harmonic functions,
and the Poisson integral formula.
Contents
Lecture Log and
homework assignments
- Instructor: Prof. Jeremy Tyson
- Office Location: 330 Illini Hall
- Office Phone: 244-4132
- Email:
tyson@math.uiuc.edu
- Office Hours: Wed 2:00-3:00, Thu 10:00-11:00
- Lecture Times: MonWedFri 11:00-11:50
- Lecture Location: 445 Altgeld Hall
- Course web site:
http://www.math.uiuc.edu/~tyson/542f07.html.
- Textbook: An Introduction to Complex Function Theory
by Bruce Palka (Springer, 1991).
- Other recommended texts (on reserve at the Math Library):
- Functions of one complex variable I by J. B. Conway, 2nd
edition (Springer, 1978).
- Complex analysis by L. V. Ahlfors (McGraw-Hill, 1978).
- Potential theory in the complex plane by T. Ransford
(London Math Society, 1995).
- Princeton Lectures in Analysis II: Complex analysis
by E. M. Stein and R. Shakarchi (Princeton University Press, 2003).
- Internet resources
- Homework: There will be weekly homework assignments.
Problems will mostly be taken from the textbook, although I will
assign some outside problems as well. The grader will grade selected
problems from each assignment. We will discuss solutions to some of
the homework problems during class.
- Exams: There will be one midterm exam (date to be
determined) and a final exam ( 8-11am, Wednesday, December 12).
- Grading Policy: Grades will be computed according to the
following percentages:
| Homework |
35% |
| Midterm Exam |
25% |
| Final |
40% |
- The date for the midterm exam will be announced well in advance.
If you have any conflict with the dates of either the midterm or final
exam, contact me as soon as possible. Travel plans are not a
valid excuse for missing the final exam. Both midterm and final exam
will be closed-book, closed-notes. You will be asked to reproduce the
proofs of some important theorems on the exams, chosen from a
prespecified list (to be handed out later).
Tentative Syllabus
- Complex numbers, topology of the complex plane and the Riemann
sphere, stereographic projection, basic conformal mappings
- Cauchy-Riemann equations, analytic functions, meromorphic functions
- Contour integration, local Cauchy theory, Cauchy integral
theorem, uniform convergence, Taylor series
- Liouville's theorem, maximum modulus theorem, Schwarz Lemma,
automorphisms of the disc, Mobius transformations, cross ratio
- Laurent series, classification of isolated singularities
- Residue theory and applications, argument principle, Rouche's
theorem, winding number, general Cauchy theorem and integral formula,
simple connectivity
- Arzela-Ascoli theorem, normal families, Montel and Hurwitz
theorems, Riemann mapping theorem
- infinite products, entire functions, Weierstrass factorization
theorem, analytic continuation, introduction to Riemann surfaces
- Runge's approximation theorem
- harmonic functions, Laplace equation, analytic completion,
Dirichlet problem, Poisson integral formula