Syllabus for Math 5583 --- Fall 2005

Description:

Complex analysis studies complex functions w = f(z) where both the input z and the output w are complex numbers. On the one hand, this is similar to ordinary one-variable calculus where the objects of study are real functions y = f(x). Many of the ideas from calculus, such as derivatives, anti-derivatives and infinite series, will have analogues for complex functions. On the other hand, a complex number z = x + i y can be visualized as a point in the plane with Cartesian coordinates (x,y). Similarly for w = u + i v. So a complex function is really a pair of functions u(x,y), v(x,y) each of which depends on two real variables. Multivariable calculus concepts like partial derivatives and line integrals will play an important role. The rich geometry of the complex plane as compared to the real line gives the subject much of its special character. In addition to being one of the most beautiful subjects in mathematics, complex analysis has many unexpected applications both to other parts of mathematics and to physics and engineering.

Text:

Complex Analysis, by Stewart and Tall. This book does of good job of explaining how the major theorems derive from the underlying geometry and topology of the plane. In addition it covers the most important mathematical consequences of the theory and touches on some of the applications. Some additional applications will be covered in the lecture.

Some other good books (not required): Visual Complex Analysis, Needham (on reserve in math library), Complex Variables, Churchill and Brown (also on reserve), Complex Variables, Fisher (Dover book), Elements of the Theory of Functions, Knopp (Dover), Complex Analysis, Ahlfors (classic but more advanced).