Syllabus for Math 131, Complex Analysis
Harvard University, Fall 2003
September 15, 2003
Science Center 420
Office hours Tuesday, 12 noon--1 PM, Thursday, 2:30--3:30PM, and by appointment.
Introduction to the course
In this course, we will learn about one-variable complex analysis.
This is the study of complex functions of one
complex variable. It is simpler and more elegant than the study of
functions of two real variables, and in some ways even simpler than the
study of functions of one real variable. For instance, a complex
function with one continuous first derivative automatically has an
infinite number of derivatives, so a whole class of pathological
examples in real analysis have no analogues in the complex case. (In
fact, more is true: such a function is automatically analytic;
that is, it has a power series expansion that converges to the
We will study functions with
complex derivatives. One key fact is that, considered as maps from
the complex plane to itself, these functions preserve angles (as long
as the derivative is not zero). As a result, they are often called
conformal maps or functions. (``Conformal'' comes from Latin
roots meaning ``same shape''.)
The real strength of complex analysis comes from properties of
integrals of complex-analytic functions along paths. In particular,
these path integrals remain unchanged when the path is perturbed; this
gives a great deal of flexibility when evaluating integrals. A
fundamental theorem, the Cauchy residue theorem, reduces the
integral of an analytic function along a closed path to counting
certain singularities in the region bounded by the path. This is
quite powerful, and, indeed, many real integrals can most
easily be done using this theorem.
We will wrap up the semester by proving the Riemann mapping theorem:
given any two regions in the plane (satisfying some mild assumptions),
there is always a conformal bijection between them. (This map is not
unique, but we will be able to completely understand the different
possibilities.) This has wide-ranging applications, from random walks
to drawing maps of the surface of the brain.
You should be familiar with real analysis (as taught in
Math 23a,b, Math 25a,b, or Math 55a,b), or alternatively have taken traditional
calculus (as taught in Math 21a,b) and be familiar with how to write proofs
(as taught in Math 101).
In addition, you should have a basic familiarity with complex numbers:
you should know how to compute (1+2i)(1-2i).
We will follow Bak and Newman's Complex Analysis (required), with
additional topics taken from Needham's Visual Complex
Analysis (optional) and elsewhere. Topics will include:
Introduction to complex analysis.
- Complex differentiability: What does it mean for a function to
have a complex derivative? An explicit differential equation (the
- Visualizing complex maps: conformality (preserving angles) and
- Integrals of complex differentiable functions and invariance under
deformation of the path of integration.
- One derivative implies infinitely many. Power series expansions
- Möbius transformations.
- Harmonic functions.
- Evaluating definite integrals by counting poles enclosed (the
Cauchy residue theorem). Applications to real integrals.
- Conformal mappings and the Riemann mapping theorem.
Assignments and grading
There will be weekly problem sets, due on Friday. Collaboration on
the problem sets is encouraged, but please write up your solutions in
your own words and, as in any academic work, credit your
collaborators. Partial credit will be given for progress towards a
solution, but you must give a complete proof to get full credit. Your
lowest problem set score will be dropped.
There will be a take-home midterm and a take-home final (pending
approval from the Registrar). At your option, you may write a 5--10
page term paper in lieu of the final. This term paper would further explore a topic
related to the course that you discuss in advance with me.
The final grade will be based on:
60% problem sets;
- 15% midterm; and
- 25% final exam/term paper.
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