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 → 1f ( 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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 Spring '09
 Math, pts, Holomorphic function, Unit disk

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