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# Math 103 Assignment 3

## For February 10

## 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 **P**rove the statement **O**r to **D**isprove
the statement. In the case of disproof, you should **A**lso **S**alvage
the statement (i.e. change it as minimally as possible and prove the altered
statement) **I**f **P**ossible. Thus these problems are called
PODASIPs. Some are easier than others.

- Let a
_{1 }> a_{2 }> a_{3} > ... be
real numbers with lim_{n->infty }a_{n }= 0. Let omega
be a primitive d-th root of unity. That is to say omega = e^{i2pi/d}
where d is a positive integer.
Show that the sum a_{1}omega + a_{2}omega^{2} +
a_{3}omega^{3} + ... converges. The alternating series
test is a special case.
- 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".
)
- There exists an open non-empty biperiodic domain and a non-constant
biperiodic function on this domain. (You will have to define "biperiodic.")
- dz = dx + i dy. (You will have to define all your terms.)

## Reading

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

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 |

James | 2:00 Wednesday |

Sonya | 9:00 |

Dan | 1:00 |

Tim | 1:30 |

Joan | 2:30 |

Terrill | 2:15 Friday |