MA530_AUG99

MA530_AUG99 - 4(20 pts Let f z = ∑ ∞ n =0 z n Show that...

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QUALIFYING EXAMINATION August 1999 MATH 530 - Prof. Bell Notation: D r ( a ) denotes the disk, { z C : | z - a | <r } . 1. (20 pts) Find the smallest integer n such that there is no z C with z 11 + z 5 +8 z + 1999 = 0 and | z |≥ n . Explain. (3 11 = 177147, 3 5 = 243, 2 11 = 2048, 2 5 = 32.) 2. (20 pts) Let f : C -{ 0 , 1 }→ C be an analytic function such that f ( z )= X -∞ a n z n for | z | > 1, where a n =1for n< 0and a n =1 /n !for n 0. Determine what type of singularity f has at 0, 1 and . 3. (20 pts) Assume that F is a one-to-one analytic mapping of the square { z : - 1 < Re z< 1 , - 1 < Im z< 1 } onto the unit disk such that F (0) = 0. Prove that F ( iz )= iF ( z ) for all z
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Unformatted text preview: . 4. (20 pts) Let f ( z ) = ∑ ∞ n =0 z n ! . Show that the radius of convergence of this power series is one. Let u denote a root of unity. Show that lim r → 1-f ( ru ) = ∞ . Let Ω = D 1 (0) ∪ D ± (1). Is there an ± > 0 and a meromorphic function F on Ω such that F = f on the unit disk? Explain. 5. (20 pts) Let F denote the family of all analytic maps f of the unit disk to itself for which f (1 / 2) = 0. Find sup f ∈F { Im f (0) } . Explain....
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