, where z is complex), dynamical systems (ever seen a
Mandlebrot set?) and many
others.
In Complex Analysis, we study complex valued functions
f(z) of the complex variable
z. The same way we do in Calculus of functions of a real variable, we
can define the derivative f'
of f as a limit. When this derivative exists,
f is called analytic. It turns out
that analytic functions have incredibly
rich properties. For instance, if a function
f is analytic, that is if f' exists, then all its successive
derivatives f'', f'''..... exist
as well! (you can think of many examples of Calculus where this is not
true. Think, for instance, of
what happens to the derivative of your displacement when you put on the
breaks in a car). A central
part of complex analysis is integration, reminiscent of path
integration in vector Calculus. Complex variable integration techniques
can yield spectacular
shortcuts to solutions of complicated Calculus problems.
Complex analytic functions have a fundamental geometric
property. If you look at
analytic functions as transformations of the plane into itself, they
preserve the angles between
intersecting curves. Hence eighteenth century mathematicians used complex functions to
solve the problem of making
geographical maps that preserved angles. This simple property,
called conformality, is central to many results in complex analysis, and
gives this field of
analysis its unique geometric flavor.
This course is a joint offering of the mathematics
departments of Smith College and
Mount Holyoke College.