8/27

Complex numbers, Fundamental Theorem of Algebra

GK 1.1-3

I’m planning to keep a pdf (and latex) set of notes in the Drive folder, updated every lecture or so. It’ll mostly just be a list of definitions and theorems, with no proofs and minimal commentary.

8/29

Complex (and real) differentiability

1.4-5, 2.2

9/3,5

Power series

3.2; also Bak and Newman 2.2-3

W/Th office hours begin this week. Please don’t hesitate to use them.

First hwk due Friday 9/6.

9/10,12

Complex integration and Cauchy’s Theorem

2.1, 2.3-2.4

I encourage all of you to attend both the graduate student seminar and dept colloquia whenever possible. You’ll be lost much of the time, but it will gradually expand your view of math.

Note for technical reasons I’ve replaced the hwk submission form in Drive with a document containing a link to the form.

9/17,19

9/24

Cauchy’s integral formula and first consequences.

Winding numbers

2.4-5, 3.1-3.6

See defn 4.4.4 and Lem 4.5.5 in GK.

9/26

Isolated singularities of holomorphic functions

GK 4.1

9/29, 10/1

Laurent series

GK 4.2-4

In my discussion of simply connected domains, I will largely avoid the standard characterization—i.e. that all closed curves in the domain are homotopic to points.

10/3,8

Cauchy’s Theorem on general domains, the Residue Theorem, and computation of real integals

GK 11.2

GK 4.5-6

I will hold off proving the general version of Cauchy’s Theorem til next week.

10/10

Cauchy’s Theorem and simple connectivity

GK 11.1-2,4

Note that the midterm will take place Wed, October 16 from 5:30-7:30 in 258 Hurley (across from the math office).

10/15

The argument principle and consequences

GK 5.1-5.3

No Wed office hour this week. Instead I’ll be available after class Tues 10/15 til 1:30 PM.

10/17,29

Riemann sphere, rational functions

GK 4.7

10/31, 11/4-6

Moebius transformations

Automorphisms of the disk and plane

GK 6.3

GK 6.1, 6.2

11/8

The Poincare metric

See e.g. Zakeri 4.4 and 4.5

This topic isn’t covered in GK. Of all the things I learned *after* my graduate complex analysis course, this is the one I wished most that I’d learned sooner.

11/12

Conformal maps between domains: examples.

There are several Thursday colloquia coming up, and I’d like to attend them, so I’ll need to move my Thursday office hours. This week, I’ll hold them right after class (i.e. 12:30-1:30). In weeks to come I’ll likely start them a little after 5 PM when the colloquium ends. I’ll post any updates below on this page.

11/12-14

The Riemann Mapping Theorem

GK 6.4, 6.5, 6.7

GK use the term ‘holomorphically simply connected’. For us, that just means ‘simply connected.’ The difference is that GK doesn’t discuss the latter notion til later in their book.

11/19

Montel’s Theorem

GK 6.5 (Arzela-Ascoli Thm stated in appendix A)

Note that I’ve updated hwk 10, extending the deadline til 11/22 but also reassigning one of the probs from hwk 9.

11/21-12/3

Harmonic functions

GK 7.1-7.6

Office hrs 5-6 on 11/21. Also after class on 11/26.

I’ll post hwk 11 (the last one) on 11/30. I haven’t yet decided on a deadline--sometime (e.g maybe Wed 12/11) during the last week of class.

12/5

Reflection principle, and other applications of harmonic functions to complex analysis

GK 7.5, 7.9

12/10

Dirichlet problem on more general plane domains

GK 7.7-7.8

Instead of my usual Wed and Thurs office hours this week, I will be available to talk hwk, etc from 5-6 PM on Tuesday. Keep in mind the last hwk is due Wed 12/11.

12/12

Review

I’d esp like to spend this lecture going over some old hwk problems—esp ones that gave people more trouble.

I’ll be available in my office Tuesday Dec 17 starting 3 PM (til 4 or later).

Final Exam Wed Dec 18, 4:15-6:15 PM in 231 Hayes-Healy (our usual lecture room)