Math 103: Complex Analysis, Fall 2007

Plan for week 4 (Meeting Thursday, Sep. 27)

Reading

Chapter III sections 1—5. On the face of it, most of this week's work is about real functions of two variables. It starts with some things you should have learned in several variable calculus and ends with things you might discuss in a PDE class. We know from week 3, though, that we are really setting the foundations of complex analysis, and we will cash in on our work beginning in week 5. Green's theorem, the theorem on harmonic conjugates, and the mean value property are all Theorems.

Presentations

I am now asking all presenters to prepare notes including exactly what will go on the boards during their talks. (You can, of course, have more than this in your notes.) Your notes should be good enough that you need have nothing else in your hands (other than chalk) when you give your talks. As always, be sure to try to follow the advice outlined in my advice on presentations.

Corey

Harmony again. In particular, the theorem in section 3 is important. Whatever else you do, you should make sure that we understand what that theorem means and why it is true.

Ben

The mean value theorem. You could give the proof, or you could show us some of the consequences.

Common Problems

III.1:7, 8.
III.2:3,4.
III.3:1, 2
III.4:1, 3.
III.5:1, 3, 5 or 6, 8.

Here is one more problem. Give an example of a complex valued function which is analytic in the open unit disk, extends continuously to the closed unit disk minus one point, has modulus one on those points of the unit circle for which it is defined, but is nonetheless unbounded.

Notes: In III.3.1, you should be able to do these with little calculation and no integration.

As usual, consider these to be the highest priority problems, and depending on your time and abilities consider doing more, or easier, or harder (or all three) problems.

Food

Food will be provided by Rebecca.