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homework108c3 - C such that f z = Cg z What if g does have...

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HOMEWORK 3 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, APRIL 21, 2009 (1) Let f ( z ) = z be defined on R > 0 . Determine the radius of convergence of its powerseries around 1, both by explicitely considering the series, and by general theory about holomorphic functions. Consider the analytic “exten- sion” of f ( z ) to U = C \ R 0 as given in the first problem set. Determine the radius of convergence of the powerseries around - 4 + 3 i . Why doesn’t this series converge to f ( - 4 - i ) at z = - 4 - i ? (2) Suppose f and g are entire functions (i.e. holomorphic on C ) and g never vanishes. If | f ( z ) | ≤ | g ( z ) | for all z , prove that there exists a constant
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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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