# Math 103 Assignment 3

## Common Homework

In section 4 of chapter IV of Palka, do the problems 1, 4, 7, 9, 12, 18, 27, 28. Look over the problems for the parenthesized reading as well.

Also, here are four less straightforward problems, which do not necessarily relate directly to this week's reading, but which arose in our discussion. In each case, I will make a statement which is probably not entirely true. Your job is to Prove the statement Or to Disprove the statement. In the case of disproof, you should Also Salvage the statement (i.e. change it as minimally as possible and prove the altered statement) If Possible. Thus these problems are called PODASIPs. Some are easier than others.

1. Let a1 > a2 > a3 > ... be real numbers with limn->infty an = 0. Let omega be a primitive d-th root of unity. That is to say omega = ei2pi/d where d is a positive integer. Show that the sum a1omega + a2omega2 + a3omega3 + ... converges. The alternating series test is a special case.
2. If f is analytic in a domain, D, then f is conformal in the region D, in the sense that f preserves angles. (You will have to define "conformal". )
3. There exists an open non-empty biperiodic domain and a non-constant biperiodic function on this domain. (You will have to define "biperiodic.")
4. dz = dx + i dy. (You will have to define all your terms.)

Read Chapter IV and section III.5. Sections III.5 and IV.3 are interesting and of some relevance to people who learned some old fashioned notation (III.5) and some more sophisticated integration (IV.3). I would advise that you read all four sections in order, but when it comes time to concentrate, concentrate on the middle two sections first.

## Presentations

• Tim: IV.1
• James: IV.2

In both cases, be prepared--at least--to tell me what you think is the most important thing in your section when you come to speak to me Thursday. I will not give any further suggestions in public, but if either of you feel in need of direction, feel free to talk to me at any time.

## Food

Dan will bring the food for break.

## Thursday Conferences

Student Time
James2:00 Wednesday
Sonya9:00
Dan1:00
Tim1:30
Joan2:30
Terrill2:15 Friday