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MATH 440/508 - Assignment #5 Due on Friday December 3, 2010 Name —————————————– Student number ————————— 1 Problem 1: Prove that an entire function which is not a polynomial takes on every complex value but one infinitely often. 2 Problem 2: Suppose r, r0 ∈ (0, 1) and h is an analytic isomorphism between the annuli T = {z; r < |z| < 1} and T 0 = {z; r0 < |z| < 1} . Suppose also 0 that h extends to a continuous map from T to T which takes the unit circle to itself and the circle of radius r to the circle of radius r0 . Show that h can be extended to an analytic automorphism of the open unit disk D that takes 0 to 0. Conclude that r = r0 . 3 Problem 3: Assume φ is an analytic function from D = {0 < |z| < r} to C\{0, 1}. Consider the sequence of analytic functions fn (z) = φ(z/n) on D for n ∈ N. Prove that fn converges normally on D to either ∞ or an analytic function f on D. Use this to prove that 0 is not an essential singularity for φ. (This proves the Big Picard theorem.)