542 MAT Complex Analysis I
Problem Set 7
due Tuesday, March 27
Problem 1. (i) Let a be real, 0 a < 1. Let Ua be the open set obtained from the unit disk {|z| < 1} by removing the segment [a, 1] of the real line. Construct a conformal isomorphism between U0 and Ua for a > 0. (ii) Construct a conformal isomorphism between U0 and the unit disk {|z| < 1}. Problem 2. Let f : D D be a holomorphic map of the unit disk D = {|z| < 1} into itself. Show that for all a D |f (a)| 1 . 1 |f (a)|2 1 |a|2 Hint: Let g be an automorphism of D that maps 0 to a, and h an automorphism that maps f (a) to 0. Apply the Schwarz lemma to F = h f g. Problem 3. (Blaschke products) A function of the form B(z) = z z1 1 z1 z z zn 1 zn z ,
where || = 1 and |zj | < 1 for all 1 j n, is called a nite Blaschke product. Check that B(z) is holomorphic in the closed unit disk {|z| 1}, maps this disk into itself and has zeroes at z1 , . . . , zn . Suppose that f is holomorphic in the closed unit disk {|z| 1} and maps this disk into itself. Suppose also that |f (z)| = 1 whenever |z| = 1. Show that f must be a nite Blaschke product. Problem 4. Let f : D D be a holomorphic map of the unit disk D = {|z| < 1} into itself such that f (0) = f (1/2) f = (1/2) = 0. Show that f 1 4 1 . 21
Show that the upper bound 1/21 cannot be improved. Hint: Use Blaschke products. Note: I saw problems like this very often on various comprehensive/qualifying exams. Problem 5. Prove that the sum of the series
g(z) =
k=1
(1)k z+k
denes a meromorphic function in C, and identify its poles. Can a formula for g (z) be obtained by dierentiating the series term-by-term? (give proof either way).
Problem 6. Let U be an open set. Suppose the functions fn are holomorphic in U and converge to f uniformly on any compact subset of U ; assume that f is not identically zero. (i) Let S = {z U : fn (z) = 0 for some n}, i.e S is the set of all zeroes of all functions fn . Show that the zeroes of f in U are identical wi...