Jim Kelliher

Real and Complex Analysis, Third Edition by the master, Walter Rudin

The following are my recommendations for selected exercises to work from Rudin. They are heavily biased by the exercises I have worked myself. I indicate those cases in which I have not worked the exercise myself, or one essentially equivalent to it,

Chapter 1:

All. Exercise 7 is useful in solving a lot of later exercises.

Chapter 2:

1-2, 5-9, 10 (using measure theory!), 11-17, 20-25

Chapter 3:

1-14, 16, 17, 25

Exercise 17 appeared in slightly different form on the January 2002 prelim.

Chapter 4:

If you have only the summer to prepare and aren't already ahead of the game, I would recommend not doing any problems from this chapter, since the material often does not show up on the prelim (throw the dice!). However, if you have the time, I would recommend

1, 2, 5, 9, 13, 14, 16 (I have not worked 14 or 16).

Chapter 5:

Even more so than Chapter 4, given limited time, work no exercises from this chapter, though you should still read both chapters 4 and 5 (or equivalent material), since chapters 9 and 10 use results from them. I have no recommendation for exercises.

Chapter 6:

1, 2, 4, 5, 9, 10 (a)-(e)

Chapter 7:

1-6, 8, 10, 14, 17, 19, 20

("Lip 1" of exercise 10 is defined in exercise 11 of Chapter 5.)

Some prelims almost ignore differentiation, but some key on it, so trying to get by without understanding it is really rolling the dice. Wheeden & Zygmund's presentation of it should also be studied, especially their theorems regarding absolutely continuous functions and functions of bounded variation. Exercise 13, which I did not recommend, includes a number of these results, but the presentation in Wheeden and Zygmund is more coherent.

Chapter 8:

1, 2, 3, 4, 9, 16

Chapter 9:

In theory this material is not on the prelim syllabus, though it has shown up on a few past prelims, so maybe the syllabus was changed. In any case, some familiarity with this material is useful. In particular, Remark 9.3(a) and Theorem 9.13 are worth remembering if nothing else.

Chapter 10:

1-22

Chapter 11:

I have worked none of these problems, so I have no recommendations. I am not sure why I chose not to work them, but it is probably because questions on harmonic functions seldom show up on the prelim, so I gave it low priority. But Dr. Rosenthal this past semester (Spring 2002) dwelled on harmonic functions, and they are on the prelim syllabus, so perhaps it would be worth working a few of the problems.

Chapter 12:

1-8

Chapter 13:

1

The topic of this chapter is not on the syllabus, but the first exercise is a classic that one should know how to work. You might also want to show that Moebius transformations are the only bijective meromorphisms from the Riemann sphere to the Riemann sphere.

Chapter 14:

1-5, 7, 10, 30, 31

Exercises 30 is roughly equivalent to Conway's description in Chapter III Section 3 of Moebius transformations, and exercise 31 is roughly equivalent to Conway's exercises 20-27 of that same section.

Chapter 15:

1-4

I recommend only exercises 1 through 4 because they are the only exercises from this chapter that I worked. The topics in this chapter are on the syllabus, but seldom appear; you can decide whether to make the same choice I made.

Chapter 16:

1, 3, 5-10

Exercise 1 was on a prelim, exercise 8 and minor variations of it have been on several prelims (probably the most repeated question ever). I include exercise 10 because I have a wild hunch that it will be on this summer's prelim. I have not worked all of these problems.

Chapter 17-20:

Not on the prelim syllabus, at least mostly not.