Home page for Math 503, fall 2007

About the course | The instructors | The text and other references | Students in the course | Outline diary of the course
Part 2
Part 1

See the award-winning YouTube video
by Douglas Arnold and Jonathan Rogness:
Möbius Transformations Revealed!
Their Riemann sphere is on top of C. But
more than one million (1,000,000) views!

Things to do
Final letter grades were submitted to the Registrar's computer late yesterday (Friday, December 21). Here is information about grading. Here's one last esthetic comment.

This one or this one?

The final exam grade of students who analyzed Aut(C*) (problem 5) using a theorem related to the name of the character in this picture has been decreased.
  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?
Again, this is similar to using Lebesgue Dominated Convergence for situations where the functions converge uniformly: killing a mosquito with a sledgehammer.

Other class material

(with PDF links)
What is it?Handed out
or posted
The final exam The final as it was given (with the word "Preparation" deleted from the title, though). Grades ranged from 20.5 to 58.9 out of 60. Since part of the purpose of the final was to prepare our first-year grad students for the written exams, I'll return the exams to these students in their department mailboxes "soon".
This has been done (1/4/2007).
Final exam information
Updated 12/14/2007
See these problems to help prepare for the final exam.
Mr. Williams kindly informed me that the diagram drawn for problem 6 was mislabeled. This was unintentional, and this copy has been changed (i to –i and 1+i to 1–i). I regret the confusion.
A result from the mid-20th century This is a discussion of Hans Lewy's example of a very simple linear partial differential equation with no solution. It relies on some easy (!) complex variables facts. 12/11/2007
Mittag-Leffler & Weierstrass Some notes about constructing holomorphic and meromorphic functions with specified behavior (Mittag-Leffler and Weierstrass Factorization Theorems). We only had time to discuss some special cases, so here is a fairly direct and naive approach to these results, together with some neat corollaries. 12/11/2007
Problem set 6 Please hand in solutions on Tuesday, December 4.
The evidence mentioned in problem 6 is here: k=4 and k=41.

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?

The midterm exam The midterm as it was given. Grades ranged from 20 to 48 out of 50. 11/15/2007
Problem set 5 and midterm exam information
Updated 11/7/2007
Further updated 11/9/2007
Please hand in solutions on Tuesday, November 13, and prepare for an exam on the same day.

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.

& 11/7/2007
& 11/9/2007
Problem set 4 Please hand in solutions on Tuesday, October 23. 10/14/2007
Problem set 3 Please hand in solutions on Tuesday, October 9. Please see the currently posted version, with a change made in response to some student observations. 9/28/2007
Problem set 2 Please hand in solutions on Friday, September 28.
In order to avoid temporal contradications, I have changed to due date, following the suggestion of Mr. Amos.
Problem set 1 Please hand in solutions on Friday, September 15. 9/9/2007
Problem set 0 Please hand in solutions on Friday, September 7. 9/2/2007
Information sheet A sheet to be passed out on the first day of class. 9/1/2004
Math 503 from the past
Links to webpages and course material for the last two times I've taught Math 503.
Fall 2004     Fall 1997

Maintained by greenfie@math.rutgers.edu and last modified 9/2/2004.