MATH 227 Final, Spring 2004

Math 227 — Final Exam (Spring 04) B. Shayva Problem 1 (9 pts each) Use complex integration to evaluate the following integrals. co x dz j i) _ (i I l+ct de a I 5 ~ 4cosz Problem 2 (a) (12 pts) Prove that if u is harmonic in a star-shaped set X, then u = Ref for some f € O(.X). (b) (5 pts) State and prove Liouville’s theorem for harmonic functions. (c) (15 pts) Let y be harmonic in the punctured plane z ¥ 0. If y is positive there, prove that ¢ is a constant. Problem 3 Let - fle) = 30 Anz” n=0 be holomorphic in the open disc jz| < 1. Also, let the point z = 1 be an isolated, but not an essential singularity of f(z). (a} (17 pts) What, beyond not being essential, is true of the singularity if Ay > 0 as n + co ? (b) (8 pts) What if |.4,} < 1 for all n ? Problem 4 Suppose f(z) = u(z) +iv(z) is an entire function with v(0) = 0. (a) (9 pts) Write f(z) = Ao + O32, 2Anz”. Show that if R > 0, then nS 1 27.R” Qa An = | ul Re!®)e7i" ag (n =0,1,...). . 0 (b) (6 pts) Show that if |z| < R, then Qa 10 4 y fle) = bP PEE re) a0. Oa lo Re? —z (c) (10 pts) Suppose there exist A, B,a > 0 such that lu(z)| < A+ Bf2z|*. Show that there exist C, D > 0 such that \f(2)| < C+ Dizi*.