Home page for Math 609: Functions of a Complex Variable
Instructor: Charles L. Epstein
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The Course
Functions of a complex variable has a special place in mathematics: it
is the
subject where things work out as well as can be expected and it is also
the
birthplace of many fields of modern mathematics. For example
topology, Riemannian geometry, algebraic topology, partial differential
equations, singular integral operators, pseudodifferential operators,
index theory, much of modern algebra, analytic number theory, harmonic
analysis
all trace their roots to the study of functions of a complex variable.
This
semester we will use E. Stein and R. Shakarchi's Complex
Analysis as the text, but we will not follow it slavishly.
We
will take a very analytic approach. This will be a graduate level
course, we will not spend a great deal of time on the elementary parts
of
the subject, and will take advantage of the real analysis you studied
in
Math 608.
Problem sets
will be
assigned every week on Tuesday, due the following week on Tuesday. I
very much prefer that students do the
problem sets
alone. We may have a
take-home midterm and final exams.
- The class meets on TTh from 1:30-3:00.
- My office hour is Mondays 3:30-5PM. Contact me by e-mail for an
appointment if you can not come during this time.
- My office in the Math Department is 4E7 DRL, tel. 8-8476.
- email: cle@math.upenn.edu.
Send e-mail if you have a question or need to contact me.
Syllabus
- Elementary theory of holomorphic functions: Applications of
Cauchy's theorem
- The residue theorem and the winding number; Toeplitz operators
and the Szego index theorem
- Meromorphic functions, covering spaces and monodromy
- The Cauchy-Riemann Equations and Runge's theorem
- The Fourier transform, and the Paley-Wiener theorems
- Methods of asymptotic analysis: Laplace's method and
stationary phase
- The theory of entire functions
- The Mittag-Leffler and Weierstrass theorems
- Conformal mapping
- Elliptic functions
Announcements
Problem sets
- Problem set 1, due January 24, 2008.
- Problem set 2, due
January 31, 2008.
- Problem set 3, due
February 7, 2008.
- Problem set 4, due
February 14, 2008.
- Problem set 5, due
February 21, 2008.
- Problem set 6, due
March 6, 2008.
- Problem set 7, due
March 28, 2008.
- Problem set 8, due
April 10, 2008.
- Problem set 9, due
April 30, 2008.
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