 Math  502,  Spring 2006 ## Complex Analysis

This is the second part of the basic graduate sequences course in analysis.  You can find the lecture notes for the first part here.

In this course we will study the behavior of differentiable complex-valued functions f(z) of a complex variable z.  The key idea in the course is that complex differentiability is a much more restrictive condition than real differentiability.  In fact, complex-differentiable functions are so rigid that the entire behavior of such a function is completely determined if you know its values even on a tiny open set.  One understands these rigidity properties by making use of contour integration - integration along a path in the complex plane.

The theory gains its force because there are plenty of interesting functions to which it applies.  All the usual functions - polynomials, rational functions, exponential, trigonometric functions, and so on - are differentiable in the complex sense.  Very often, complex analysis provides the solution to "real variable" problems involving these functions; as someone said, "The shortest path between two real points often passes through the complex domain." Moreover, complex analysis is a key tool for understanding other "higher transcendental functions" such as the Gamma function, the Zeta function, and the elliptic functions, which are important in number theory and many other parts of mathematics.  A secondary aim of this course is to introduce you to some of these functions.

One of the surprises of complex analysis is the role that topology plays.   Simple questions like "do I choose the positive or negative sign with the square root" turn out to have surprisingly subtle answers, rooted in the notion of the fundamental group of a topological space (which you will be looking at in the Topology and Geometry course parallel to this).  These topological notions eventually culminate in the notion of a Riemann surface as the correct global context for complex analysis.  We will not develop this idea fully, but we will discuss `multiple-valued functions' and their branch points; again, we will try to illustrate how these exotic-sounding concepts help in doing practical calculations.

We will be using ANGEL (Penn State's course management software) for this course and you will need to log on to ANGEL regularly throughout the semester.   You can connect to the ANGEL main page by pressing the button below.  You will find the course syllabus and other relevant information on the ANGEL page for the  course. Some course items will be linked directly from this page; they will all be available through ANGEL also.

·         Lecture notes

·         Homework 1

·         Homework 2

·         Homework 3

·         Homework 4

·         Homework 5

·         Homework 6

·         Homework 7

·         Homework 8

·         Homework 9

·         Homework 10

·         Homework 11

·         Homework 12

·         Solutions to in-class exercises.

·         Lecture notes (full set) reformatted for easier printing.

INFORMATION ABOUT THE MIDTERM: Here is a copy of the midterm exam that I used last time I taught this course. The midterm this time will have a similar format: 4 questions, each consisting of a 'theory part' and a 'practical part'. Only two questions need be completed for full credit.  