PDEs and complex analysis

To students of AM 105a:

On 26 October 2007, Albert Tsou sent in the following Q & A.

Q: Why did the mathematician name his dog "Cauchy"?
A: Because the dog left a residue at every pole.

As you know, certain sections in Saff and Snider upset me, and I have deviated from the book. We all agree that no text should be held as an authority. But now, both you and I are paying for the deviation.

The notes here follow closely my lectures. Several students have mentioned that their own notes may reproduce what was written on the blackboards, but not what I said in class, so that they found it hard to reproduce the ideas.

The notes, as well as the corresponding lectures, attempt to teach a particular application of complex analysis: using an analytic function to solve a partial differential equation. The primary example used here is electrostatics. I have several objects in mind:

• Link math to a physical problem. To make this link, you need to review the basics of electrostatics, to the extent that you can make physical sense of the mathematical results.
• Link PDE to analytic function. I use an approach to show that this link can be made for PDEs other than the Laplace equation. I learned the approach from technical papers when I was a graduate student here in late 80s, and used it in my papers on interfacial fracture mechanics. I have never seen this approach in any standard textbooks.
• Show techniques of using complex analysis to solve a PDE.

Thus, the “applied” content of these notes is somewhat more than the corresponding sections in Saff and Snider. One way to learn this material is to work through these notes, adding your own notes and inventing your own exercises along the way. For example, I don’t have time to draw figures using WORD. Please sketch your own figures from the notes you took in class. Fill the 3 inch margin on the right side.