## HOMEWORK 5 DUE APRIL 10 1. Let B (0; 1) be the open unit disk. Find a (nonzero)

HOMEWORK 5 DUE APRIL 10 1. Let B (0; 1) be the open unit disk. Find a (nonzero) holomorphic function in B (0; 1) which has innitely many zeros in B (0; 1). 2. Let D := Cn f??1; 1; ig and f : D ! C holomorphic such that f (0) = 0: Determine f; given that jfj (z) jlog j1 ?? zjj + jlog j1 + zjj + jlog jz ?? ijj ; for all z 2 D: 3. For a region ; we denote Aut () the set of all bijective and holomorphic maps f : ! . Determine (a) Aut (C) (b) Aut (Cn f0g) (c) Aut (Hn fig) ; where H =fz : Im (z) > 0g is the upper half plane. 4.
HOMEWORK 5 DUE APRIL 10 1. Let B (0; 1) be the open unit disk. Find a (nonzero) holomorphic function in B (0; 1) which has innitely many zeros in B (0; 1). 2. Let D := Cn f??1; 1; ig and f : D ! C holomorphic such that f (0) = 0: Determine f; given that jfj (z) jlog j1 ?? zjj + jlog j1 + zjj + jlog jz ?? ijj ; for all z 2 D: 3. For a region ; we denote Aut () the set of all bijective and holomorphic maps f : ! . Determine (a) Aut (C) (b) Aut (Cn f0g) (c) Aut (Hn fig) ; where H =fz : Im (z) > 0g is the upper half plane. 4. Let f : B (0; 1) ! C be a function so that f2 and f3 are both holomorphic. Prove that f is holomorphic as well. 5. Let f and g be entire functions so that jfj (z) jgj (z) for all z 2 C. Prove that there exists a constant c 2 C so that f = cg: 6. Suppose a 2 C is a pole for a function f : B (a; r) n fag ! C. What kind of singularity does ef have at z = a? 7. Let f be analytic in the punched unit disk B (0; 1) n f0g such that there exists a positive integer n with f(n) (z)1 jzjn ; for all z 2 B (0; 1) n f0g : Prove that z = 0 is a removable singularity for f. 8. Determine the set of all z so that1X k=??1akzk converges where ak = k6 3k 1 k6+1 if k 0 if k

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