8/10-8/14

Complex numbers, proof of the Fundamental Theorem of algebra;

Real vs. complex linearity and differentiability.

1.1-1.2

My notes

No regular office hours this week, but feel free to make an appt w. me.

You’ll find that my perspective and presentation are often quite different from that of Greene and Krantz. I think it’s good to have more than one point of view on things.

8/17-8/21

Holomorphic functions.

Path integrals

1.3-1.5

You’ll notice I discussed some of 1.4 and 2.2 last week.

1^st hwk due Friday 8/21 by noon.

8/24-8/28

Cauchy's Theorem and Integral Formula

2.1-2.5

My proof of Cauchy’s Theorem for rectangles and Cauchy’s Integral Formula will follow Ahlfors 4.1 and 4.2--unlike GK, I will not define holomorphic fns to be continuously differentiable, which allows for somewhat shorter proofs.

8/31-9/4

Consequences of Cauchy's Theorem

3.1-3.5

For the time being at least, I’m uploading to Google Drive outlines of my class notes before class, expanding them during lectures, sprucing them up afterward and then re-uploading the expanded outlines.

9/7-9/11

Power series and Laurent series, zeroes and isolated singularities

3.5-3.6, 4.1-4.4

This week’s homework isn’t due til Tuesday (9/15) of next week!

9/14-9/18

Residue Theorem: theory and practice

4.5-4.6

Take home midterm (Thurs-Sat 9/17-19)

9/21-9/25

meromorphic functions and the Riemann sphere

zeroes and poles

4.7, 5.1-5.2

9/28-10/2

maximum principle; linear fractional transformations

5.3-5.5, 6.1-6.3

10/5-10/9

The general form of Cauchy's Theorem

11.2-11.4

10/12-10/16

Riemann mapping theorem

6.4-6.6

10/19-10/23

Harmonic functions

7.1-7.6

10/26-10/30

More harmonic functions

11/2-11/6

Schwarz reflection and consequences.

The modular lambda function.

7.5

10.4-10.5

11/9-11/13

Big and little Picard Theorems and the strong version of Montel.

11/16-11/20

Finals week