Fair game for Exam 2:

- Types of singularities and their properties

- Residues

- More contour integration

- Use of theorems, particularly those in 3.4 "The argument principle and applications"

- Property of being simply connected

- Branches of logarithm

- Conformal equivalence and examples of conformal maps, including the LFT information not covered so much in Stein Shakarchi as in Ahlfors.

- Everything from Chapters 1 and 2 except 5.2, 5.3, and 5.5 (the parts about uniformly converging sequences of holomorphic functions)

You are allowed two sheets of notes (front and back). My idea is that you can make a new sheet to supplement your old sheet. Of course, no one will check if you have two brand new sheets. I allow a sheet of notes mainly because I suspect the act of making the sheet is more useful than the sheet itself. Secondarily, it's nice to have a quick reference for all the necessary hypotheses of your useful theorems.

Notes from before:

The second exam will involve applications of residues and more contour integration. You are also again expected to be able to apply theorems learned in class, now with emphasis on the theorems in Chapter 3. Finally, you are expected to be able to construct conformal maps fulfilling given properties.

Suggested ungraded exam study questions:

Stein-Shakarchi Ch. 8 Exercises 4, 5, 8 (only the part about finding F_i)

In Ahlfors' book:
p. 78: 2, 4, (3 optional cool problem)
p. 80: 1 (NEW)
p. 96-97 2, 3 (click for hint)

Prove, in a different way than Stein and Shakarchi's text, that Example 7 of Chapter 8.1.2 gives a conformal equivalence between the upper half disk and the upper half plane. This way is by expressing the map as the (inverse of the) composition z --> w described on page. 94 in Ahlfors' text, composed with one more map, f(z) = -z. [This problem is recommended before trying Ahlfors p. 97 #3.]