Home page for Math 503, fall 2007

This one or this one?

1. If you can prove the theorem used, come to me and do it. I will then remove the penalty.
2. Couldn't you prove what you wanted with the Casorati-Weierstrass Theorem?

The originator of question #6 is Dr. Vincent Vatter. His published analysis of the problem connected with this question used a result called Pringsheim's Theorem. This and a great deal more about the application of "simple" complex variable techniques to enumerative combinatorics is explained in a publication by Phillipe Flajolet with title Symbolic Enumerative Combinatorics and Complex Asymptotic Analysis. I think these are notes taken by Yvan Le Bourgne, who was then (bien sûr!) a grad student. Don't grad students do all the work?

A change has been made (11/7/2007):
Problem 5 on the second page was wrong as stated. Thanks to Jinwei Yang for pointing this out. The sequence {1/n}_n in N is discrete in C but not closed. The problem has been restated. In complex analysis discrete subsets arise, for example, as zero sets of non-constant holomorphic functions and are therefore (holomorphic implies continuous) always closed, so I have added the word "closed" to one part of the question and taken away the word "open" (now superfluous!) from the other part of the question.

Further changes (11/9/2007):
In problem 3, the set capital omega is now defined when it first appears. And, in the stem of the problem, I added, just for clarification, that the coefficients of the power series may depend upon both the function and the "center". Grudging thanks are given to Mr. McRae, Ms. Naqvi, and Mr. Williams.