Tentative Schedule for 403:01, spring 2002

This is a rather brisk schedule. Changes may need to be made.

Lecture SectionSection title Reality
What was actually done
1 1.1
1.1.1
Complex Numbers and the Complex Plane
A Formal View of the Complex Numbers
Meeting #1
2 1.2 Some Geometry Meeting #2
31.3 Subsets of the Plane Meetings #3 and #4
See note 1 below.
41.4 Functions and Limits Meeting #5
5, 61.5 The Exponential, Logarithm, and Trigonometric Functions Meetings #6 and #7
71.6 Line Integrals and Green's Theorem Meetings #8 and #9
See note 2 below.
8 2.1 Analytic and Harmonic Functions; the Cauchy-Riemann Equations Meetings #10 and #11
9, 10 2.2Power Series Meetings #11 and #12
11 Exam 1 Meeting #13
122.3
2.3.1
Cauchy's Theorem and Cauchy's Formula
The Cauchy-Goursat Theorem
Meeting #14
See note 3 below.
13, 14 2.4 Finish Cauchy-Goursat
Consequences of Cauchy's Formula
Meetings #15 and #16
15 2.5 Isolated Singularities Meetings #17 and #18
See note 4 below.
16, 17 2.6 The Residue Theorem and its Application to the Evaluation of Definite Integrals Meetings #19 and #20
18, 19 3.1 The Zeros of an Analytic Function Meetings #21 and #22
See note 5 below.
20 3.2 Maximum Modulus and Mean Value Postponed until later
Will be meeting #27
See note 7 below.
21 Exam 2 Meeting #23
22 3.3 Linear Fractional Transformations Meetings #24, #25,
and #26
See note 6 below.
23, 24 3.4 Conformal Mapping
25, 26 3.5 The Riemann Mapping Theorem and Schwarz-Christoffel Transformations
27 4.1Harmonic Functions
284.3 Integral Representations of Harmonic Functions

Notes to the schedule

1 The introduction to varieties of sets (open, closed, etc.) took longer than I thought. And in addition some effort was devoted to explaining why such properties might be interesting, using the idea that somehow these principles should be correct about "calculus": that derivative should be rate of change, and that a function whose derivative is always 0 should be constant. These statements were briefly investigated for functions in R (the real line) and R2, the plane, in order to provide some background for the word "connected".

2 Review and further study of line integrals and harmonic functions took longer than expected.

3 I did not cover the Cauchy-Goursat variant of Cauchy's Theorem, and will probably only mention it in passing. The result, strongly pruning the hypotheses and still getting the same conclusion, is striking, but the proof technique does not seem to be used anywhere else in the course.

4 Much more discussion of isolated singularities was given. This took more time than I had initially allocated.

5 In the second lecture nominally devoted to this topic we principally reviewed some definite integrals computed with residues (foreshadowing the second exam), and actually spent little time on Rouché's Theorem.

6 A bare statement of the Riemann Mapping Theorem was also given.

7 I don't know what happened to meeting #28 -- the class did, I believe, meet 28 times! So much for my record-keeping ability (and my counting!).

General remark
For me the schedule was quite ambitious. Perhaps I spent too much time reviewing homework. I probably could have been more "ruthless" and pushed on and on.