Unformatted text preview: C such that f ( z ) = Cg ( z ). What if g does have zeros? (3) Let φ : D (0 , 1) → D (0 , 1) be holomorphic and φ (0) = 0. Deﬁne recursively φ = φ and φ n +1 = φ ◦ φ n . Suppose lim n →∞ φ n converges uniformly on compact subsets of D (0 , 1). What can you say about the limit function? (4) Suppose f is an entire function, such that for any p ∈ C the powerseries f ( z ) = X k ≥ a k ( z-p ) k has at least one zero coeﬃcient (i.e. a k = 0 for some k ). Prove that f is a polynomial. Hint: Pigeonhole principle? 1...
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- Spring '09
- holomorphic functions, ﬁrst problem