Complex Analysis

Complex analysis (Spring 2016) is the study of calculus with complex numbers in place of real numbers. At a first glance, it may appear that there should not be much difference between calculus with real numbers and calculus with complex numbers, but nothing could be further from the truth!

Complex analysis is full of amazingly beautiful theorems that have no analogue in standard calculus. For instance, if you know the value of a real-valued function on some circle in the plane, that is not enough to determine its value at any point in the interior. Yet, for complex differentiable functions, this is sufficient, thanks to the delightful formula f(z_0)=\frac{1}{2\pi i}\int_C \frac{f(z)}{z-z_0}\, dz, known as the Cauchy Integral Formula. Using this formula and a bit more, one can turn many integrals into finite computations, which even allows for easy computations of some real integrals, such as \int_{-\infty}^\infty \frac{1}{1+x^8}\, dx.

Complex analysis can also be used to clear up some mysteries about functions of a real variable. For instance, consider the series 1+x+x^2+x^3+\cdots. This series converges when |x|<1, and it converges to \frac{1}{1-x}. That makes sense: when x=1, the function \frac{1}{1-x} is not defined, so it is reasonable that the series will also encounter problems there. But what about the series 1-x^2+x^4-x^6+\cdots? This series also converges when |x|<1, but this time it converges to \frac{1}{1+x^2}. But at x=\pm 1, \frac{1}{1+x^2} is a perfectly well-behaved function. What is going on? Well, complex numbers come to the rescue, and remind us that this function does not behave so well at x=i, and this can be used to explain the failure of convergence.

In the land of real functions, we are familiar with functions like f(x)=\sin(x), which only take on a narrow range of values (in this case, -1\le\sin(x)\le 1). It turns out that this is purely a phenomenon of real functions; all complex functions that are defined everywhere (except for constant functions) must take on a wide range of values. There are various theorems along these lines, starting from Liouville’s Theorem and progressing to the more demanding Picard’s Theorem. The Fundamental Theorem of Algebra, which says that every nonconstant polynomial over the complex numbers has a complex root, is a very easy consequence of Liouville’s Theorem.

The complex analysis class will meet 10 Tuesdays and 10 Wednesdays starting March 29 and ending June 8, with a break on May 17 and May 18.

Prerequisites: Knowledge of calculus is required.