Math 503: what students should know

Math 503: Complex Analysis, in the Fall, 1997, semester
Preface	Background	Syllabus	Homework	Links to 503 technical pages	The text

What students should know

Necessary math background

Students should know advanced calculus. The material in the Rutgers courses Math 311 and 312 is adequate; even better is Math 411, the first semester of the "baby Rudin" course. Students should be familiar with continuity, differentiability, and the (Riemann) integral. Students should know basic point-set topology, at the level of Munkres's Topology: A First Course (but see also Chapter II of the text, which reviews the needed point-set topology). Note that intricate examples of topological spaces won't be constructed or even relevant, but familiarity with arguments using simple consequences of compactness, connectedness, and some aspects of metric spaces (including completeness) will be useful. Certainly some linear algebra will be necessary, but additional algebraic background be needed only rarely. A previous (undergraduate) complex analysis course is not truly necessary, but would be helpful. As in any subject, a knowledge of the applications to other areas of what we'll cover is useful motivation. I won't be able to cover enough of that.

Students who have already had a graduate course in complex analysis should not enroll in this course. The course is not aimed at them and would be mostly a waste of their time. Their presence might distort the dynamics of the classroom for its intended audience.

Using the text

Most times that I have taught Math 503 I've stayed loyal to the text for some of the semester, and then I've strayed. I'll try hard to stick with the text this time, and not present alternate and possibly more interesting treatments of results. This resolution may not be successful.

Students should use the text vigorously. I believe it can be read and that the author has worked out a large number of examples in reasonable detail. Students should look at the textbook's problems for every topic covered in class and should know how to do most of them.

Grading

I'll rely almost totally on the results of the homework problems and exams (see below).

Exams

Since I last taught Math 503 we've changed from a purely oral qualifying exam to a system where the initial exam is written. Some of the written exam will cover complex analysis. So students should prepare for such an exam, and this course should aid that effort. I anticipate a "midterm" of several hours duration and a similar final exam. I also believe that mathematics has an important oral tradition, so some part of the final exam should be oral. We'll discuss this as the semester progresses.

The midterm lasted two hours, and was given on Friday, October 24. There were problems given to prepare for the midterm. The exam itself and some answers are available.

There were also preparation problems for the final exam, and here also the exam itself (given for two hours on Wednesday, December 17) and some answers are available.

Homework

I'll assign homework problems, taken mostly from the text. Students should write clear and correct solutions to the problems. I'll try to grade and return the problem solutions given to me in a timely fashion. Students should expect 2 to 4 problems per week.

I originally wanted to require each student to "write up" at least one problem during the semester using TeX. I've been persuaded not to require this, but I'm willing to advise and encourage students' efforts towards this goal.

Every student should make at least one oral presentation (perhaps a solution of a homework problem) in class during the semester. Again, having this as a requirement may be unrealistic, but I want to encourage both oral and written exposition of mathematics.

The assignments are textbook problems unless otherwise indicated. (For example, "FL" means the First Lecture notes.)

Date assigned	Date due	Problems
9/3/97	9/8/97	FL p.1:ii; FL p.3: ii & iii
9/10/97	9/15/97 9/17/97	1:4.28, 4.37, 4.38, 4.61, 5.23, 5.27
9/17/97	9/24/97	3:6.9, 6.12, 6.22, 6.27
9/24/97	10/1/97	3:6.48, 6.53; 4:4.17, 4.18

Pace of the course & syllabus

Our progress will depend on the clarity of the lectures and their reception by students. I'll try to stick to the text's presentation for much of the course. Here's my first pass at the schedule. Note that a semester course at Rutgers meeting twice a week has only 28 meetings, and that there should be exams and perhaps student presentations. Clearly more thinking should be done, but the course will probably move fast enough.

	Topic	Reading	Date
1	The first lecture	Notes	9/3/97 9/8/97
2	The algebra of complex numbers and some elementary functions	I	9/10/97
3	A bit about topology, especially continuity	II	9/15/97
4	Complex differentiability; the Cauchy-Riemann equations	III:1-2 & 5	9/15/97
5	Partial derivs & differentiability		9/17/97
6	Harmonic conjugates; meeting logarithm	III:3,4	9/22/97
7	Paths and integrals along paths	IV:1-2	9/24/97
8	Primitives and integrals; Green's Theorem implies ...	IV:2	9/29/97
9	Technicalities & the Goursat proof	V:1	10/1/97
10	A Cauchy Theorem	V:1	10/6/97
11	Winding number and another Cauchy Theorem	V:2	10/8/97
12	Derivatives and consequences: Morera, Schwarz reflection, Cauchy estimates, Liouville, Maximum Modulus, Schwarz Lemma, etc.	V:3	10/13/97
13		V:3	10/15/97
14	More consequences: Schwarz, Max Mod, unif conv, etc.		10/20/97
15	Branches of logarithm & arctan; their Riemann surfaces. Automorphisms of the unit disk	V:4	10/22/97
16	Homotopy, contactibility, simple connectivity: a homotopy Cauchy theorem	V:6-7	10/27/97
17	The hard part of a homology Cauchy Theorem	V:5	10/29/97
18	Discussion of real analyticity; beginnings of complex power series	VI:1	11/3/97
18	Weierstrass M-test; u.c.c. convergence	VII:1-3	11/5/97
19	Power series and Taylor series	VII:3
20	Laurent series	VII:3
21	Behavior near a point; discrete level sets; analytic continuation.	VIII:1
22	Isolated singularities	VIII:2
23	The Residue Theorem; some computations	VIII:3
24	Consequences: Rouche's Theorem, the argument principal, the open mapping theorem	VIII:3
25	Conformal mapping; automorphism groups of some domains	IX:1
26	The extended plane; Moebius mappings	VIII:4 & IX:2
27	More conformal maps	IX:2
28	Normal families	VII:4
29	The Riemann Mapping Theorem	IX:3
30	D-bar & analytic alternatives to homo{logy/topy}; Harmonic functions	??

What we really did ...

We certainly covered all of the material in the first 24 lectures listed above, and a selection of the material beyond that. More homework was given than is indicated above. We also discussed and proved the Mittag-Leffler Theorem and Weierstrass's Theorem (for the whole plane) and some of their consequences. In addition, some students gave short reports on several topics. We did not study conformal mappings or Moebius mappings enough.

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