This is a single course that is offered at both the 200- and 300-levels, and students may register for either Math 208 or Math 310. Assignments will be tailored to the level for which a student is registered.
The material in
this course is arguably the most coherent in our curriculum. Starting
from the idea of a complex number a + ib, where
i is the square root of -1, and defining the derivative of a
function of one complex variable just as one does in beginning
calculus, one is inexorably led to various remarkable theorems.
For example, if f(z) is a function with one complex
derivative, it automatically has infinitely many derivatives. This
fails completely for differentiable real-valued functions. The
function h(x)=x^(4/3) has only one derivative on
all of the real numbers. Furthermore, if one knows the value of a
function with one complex derivative on the boundary of a circle, one
knows the value at all the interior points. Again, this is
emphatically not the case for real-valued functions of a real
variable.
When the course ends we will have covered: the complex derivative,
complex integration, Taylor and Laurent series and residue theory. If
time permits we may do some topics from conformal mappings, which are
mappings preserving angle, but not length, or quasiconformal
mappings, which allow some angle distortion, but not too much. Such
mappings have recently found application in connection with research
on brain mapping.
Although Math 302 is a pre- or co-requisite for the 300-level version
of this course, 310 is not a continuation of 302 (as 303 is) but a
new beginning. The context for this course, the complex plane and
complex-valued functions of a complex variable, is a very concrete
one. If you enjoyed the concrete, geometric aspects of Math 205,
you'll like this material. If you want to see the "proper context"
for the power series you saw in Math 116, you should take this
course. If you want to either preview or reinforce the material on
convergence and uniform convergence that is studied in Math 302, this
course is a good choice. More generally, Math 208 works well as a
foundation for much of the Math 302 material.
