This is a rather brisk schedule. Changes may need to be made.
What was actually done
| Complex Numbers and the Complex Plane |
A Formal View of the Complex Numbers
|2||1.2||Some Geometry||Meeting #2|
|3||1.3||Subsets of the Plane||Meetings #3 and #4|
See note 1 below.
|4||1.4||Functions and Limits||Meeting #5|
|5, 6||1.5||The Exponential, Logarithm, and Trigonometric Functions||Meetings #6 and #7|
|7||1.6||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||2.2||Power Series||Meetings #11 and #12|
|11||Exam 1||Meeting #13|
Theorem and Cauchy's Formula|
The Cauchy-Goursat Theorem
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,|
See note 6 below.
|23, 24||3.4||Conformal Mapping|
|25, 26||3.5||The Riemann Mapping Theorem and Schwarz-Christoffel Transformations|
|28||4.3||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!).
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.