MATH 227 Final, Spring 2002

Math 227-Final Exam (Spring 02) B. Shayya Problem 1 Let f(2) be holomorphic in the ring A < [2] < B and suppose | f(2)| = |2|" there. (a) (10 pts) Find the logarithmic derivative of f. (b) (5 pts) Show that ye is an integer. Problem 2 (25 pts) Let x f(z) = So Ane" nel be holomorphic in the open dise :>| < 1. Also. let the point < = 1 be an isolated, but not an essential singularity of f(z). What, beyond not being essential. is true of the singularity if |A,| <n? Problem 3 , Let y(z) be harmonic in C — 0. (a) (15 pts) Prove that o(z) = Ref(s) + Alog:s| n with f(z) holomorphic in © — 0 (b) (10 pts) Is A nniqne ? Problem 4 Let fk be an entive fimetion such that fa(z)} > Ll on the circle [2] = 9. (a) (15 pts) Prove that the two equations bic) =44 37 and A(z) =8 A 5. have the same imunber of solutions in the dise [2| (b) (5 pts) Prove that sup |h(s)j > 10. Problem 5 (15 pts) Let p(z) = 2" +ay,o12"° Fee ey. > 1. Prove thar sup p(s)! > 1. crap) where D is the closed unit disc.