Jim Kelliher

Functions of One Complex Variable I, Second Edition by John B. Conway

UF = exercise gives a fact useful for solving many other exercises; at least read the exercise even if you don't work it.

DR = proven in Rudin if you prefer to look up the answer.

Chapters I and II: If you need to work these exercises, don't take the prelim this time around.

Chapter III: 

Section 1: 6, 7

Section 2: 10, 13, 14, 19

Section 3: 5, 7 (DR), 8,  9 (DR), 10 (DR), 20-27 (group properties of Moebius transformations)

The limited view of Moebius transformations presented in the text in Rudin is all that has been needed on past prelims, but I don't think you really get comfortable enough with Moebius transformations to confidently use them unless you learn more about them. These problems are roughly equivalent to Rudin's (long) exercise 31 of Chapter 14, while Conway's description of the cross ratio is equivalent to Rudin's (long) exercise 30 of the same chapter.

Chapter IV

Section 1: 1, 2, 6

 

Section 2: 2, 3, 4, 7 & 9 & 10 (but use any integration techniques you want)

 

Section 3: All

These are easy problems, but they point out little things you might not think of, and show elementary ways of using the techniques.

Section 4: 2, 3

 

Section 5: 1 (DR), 5-9

 

Section 6: All

These are quick and easy exercises, worth doing at least casually.

Section 7: 3-5 (UF).

Chapter V

Section1: 1, 4-6, 7-9 (UF), 10, 11, 13(a), (b) (UF). 16. 17 (prelim-like problem)

 

Section 2: 1-3, 4 (UF), 5, 12 (UF), 13 (was a prelim problem)

 

Section 3: 2, 3, 4-5 (UF), 6, 9, 10

 

Chapter VI (Max mod, Schwarz too important to skip any problems)

Section 1: All

 

Section 2: All

I don't understand 6(b). I never worked the problems in this section, actually, though I should have.