MA 471-671

Final Exam --- 1999

Do all problems and show all work.

1. True/False

2. Quickies. Answers only -- no partial credit.

3. Definitions. Give the precise definitions of each of the following.

4. Does the function F(z) = |z| have a complex derivative at z0=1= 1+0i? If so, prove it. If not, prove it.

5. First define what is meant be a function being chaotic. Then prove that the shift map is chaotic on the sequence space.

6. The Julia set. Write an essay describing the Julia sets of z2 + c. In your essay you should discuss the the geometry of these sets together with whatever you can say about orbits on the Julia set. Also discuss the Julia sets for the special cases c=0 and c=-2.