8/23

Complex numbers, Fundamental Theorem of Algebra

Bak and Newman 1.1-1.4

If you’ve not had much prior experience with complex arithmetic, let me know and I’ll point you to a video introduction to this that I made for another class. And do the practice problems!

8/25

Complex (and real) differentiability

Zakeri 1.1-1.2

8/30

More differentiability. Complex integration.

Zak 1.3-1.4

9/1

Cauchy’s Theorem

Zak 1.4

1^st hwk due Friday 9/2. I’ll retrieve it the next morning.

9/6, 9/8

Power series

B&N 2.1-2.3

I’ll be away at a meeting this week. My plan is to pre-record videos for these lectures.

2^nd hwk is posted as of 9/2 but not due til 9/16.

9/13

Cauchy’s integral formula and consequences

Zak 1.4

9/15

… more consequences, more winding numbers

Zak 1.5, 2.3

You’ll notice that my treatment of winding numbers and the generalized Cauchy’s Theorem is quite different than Zakeri’s. Indeed I’m following more closely the presentation from Chapter 11 of Greene and Krantz

9/20

Cauchy’s Theorem on general domains

Zak 2.4,2.5

9/22

Isolated singularities of holomorphic functions

Zak 3.1

9/27

Laurent series and the
Residue Theorem

Zak 3.3, 3.4

9/29

Applications of the Residue Theorem, meromorphic functions

Zak 3.4, 3.5

10/4

Riemann sphere, rational functions

Zak 3.2

I’ve posted a compilation (notes.pdf) of theorems, definitions, etc, in the Drive folder. Comments welcome! The tex source is there, too, in case you’d like to mess with it.

10/6

Moebius transformations

Zak 4.1, 4.3

10/11

Midterm review; more Moebius transformations.

Midterm exam 6-8 PM in HH 117

10/13

Automorphisms of the disk and plane

Zak 4.2

10/25

The Poincare metric

Zak 4.4 &esp 4.5

There will be no office hour on Wed 10/28. Thursday’s office hour will take place as usual.

10/27

Conformal maps; normal families

Zak 5.1-5.2

11/1

The Riemann Mapping Theorem

Zak 6.1

11/3

More Riemann Mapping

11/8

Harmonic functions

Zak 7.1

The current hwk is short but tricky in places. Try & get an early start so you have time to work through those.

11/10

Poisson Integral Formula and consequences

Zak 7.2-7.3

11/15

Harnack’s inequality, Schwarz reflection

11/17

Subharmonic functions

This topic isn’t covered in Zakeri. See the file “subharmonic.pdf” in Drive for this lecture and the next few. Also, “book8.pdf” in the “shaw book” subfolder.

11/22

Subharmonic barriers and Perron’s Theorem

11/29

Proof of Perron’s Theorem. Riemann Mapping Theorem revisited.

12/1

Doubly-connected domains

12/6

Schwarz-Christoffel formula.

Zak 10.3

Finally back to the textbook. Last hwk due today.

12/8

Quick tour of complex dynamics

Office hrs Thursday December 15, 12-2 PM

Final exam Friday December 16, 10:30 – 12:30 in HH 229