Although the course has no formal prerequisites, and in particular
no previous knowledge of complex numbers or complex functions will be
assumed, students should have had some prior exposure to a proof based
analysis or calculus course on the level of Math 3610. In particular,
students should have had some exposure to uniform convergence and a
proof of Taylor's Theorem.
Alternative Courses
Advanced undergraduate students who have taken a course such as Math 3610,
particularly those students planning to do graduate work in mathematics
or the sciences, are welcome to enroll in Math 5410. However,
undergraduate students with no prior exposure to complex functions may want
to consider Math 4520, the undergraduate course, as an alternative.
Math 4520 puts less of an emphasis on proofs of the major theorems
in complex analysis and puts more emphasis on what complex analysis
can be used for.
Course Description
The course will cover the basics of functions of one complex variable.
The emphasis on the course will be proofs of the foundations of complex
analysis. However, there will also be some discussion of applications
to fields such as electrostatics and fluid dynamics.
The course is also aimed at preparing mathematics graduate students
for the departmental exam in complex analysis.
Topics will include: an introduction to the complex numbers,
polar coordinates, stereographic projection, complex functions defined
as power series, the elementary transcendental functions, fractional
linear transformations, complex derivatives, the Cauchy-Riemann equations,
complex integration, the Cauchy Integral Formula and Cauchy's Theorem,
the Calculus of Residues, conformal mappings and their
applications, and perhaps some discussion of harmonic functions.
Geometric intuition is a very important part of complex function theory,
and topics will be presented from the geometric viewpoint whenever possible.
The section on Cauchy's Theorem will include an introduction to the
topology of the plane, but no prior knowledge in topology will be assumed.
Textbook
Bruce Palka,
An Introduction to Complex Function Theory, Springer-Verlag, 1995
Course Requirements
The course will have two "in-class" mid-term exams, a "take-home" mid-term
exam, and a final exam which will have both a "take-home" and "in class"
component. Students may also be asked to occasionally present
solutions to homework problems. The relative weight of each of these
in determining final grades wil be:
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