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math185f08-hw7 - f C → C be an entire function(a Suppose...

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 7 For any S C , recall that ¯ S denotes its closure and ∂S denotes its boundary. For any real or complex-valued function f , we will write f k ( z ) := [ f ( z )] k for k N . 1. Let Ω be a region and D (0 , 1) Ω. Let f : Ω C be analytic. (a) Show that 2 π Z 2 π 0 f ( e ) cos 2 θ 2 = 2 f (0) + f 0 (0) . (b) Show that for n N , n ! π Z 2 π 0 [Re f ( e )] e - inθ = f ( n ) (0) . 2. Let Ω be a region and D (0 , 1) Ω. Let f : Ω C be analytic and Γ be the curve z : [0 , 1] C , z ( t ) = e 2 πit . For this problem, we will write u ( z ) := Re f ( z ) and v ( z ) := Im f ( z ). (a) Suppose u 2 (0) = v 2 (0). Show that Z Γ u 2 ( z ) z dz = Z Γ v 2 ( z ) z dz. (b) Suppose u (0) = v (0) = 0. Show that Z Γ u 2 ( z ) v 2 ( z ) z dz = 1 6 Z Γ u 4 ( z ) + v 4 ( z ) z dz. 3. Let f : D (0 , 1) C be analytic. (a) Suppose f ( z ) D (0 , 1) for all z D (0 , 1). Show that | f 0 (0) | ≤ 1. (b) Suppose | f ( z 2 ) | ≥ | f ( z ) | for all z D (0 , 1). Show that f is constant on D (0 , 1). 4. Let
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Unformatted text preview: f : C → C be an entire function. (a) Suppose lim | z |→∞ f ( z ) z = 0 . Show that f is a constant function. (b) Suppose f (0) = 3 + 4 i and | f ( z ) | ≤ 5 for all z ∈ D (0 , 1). What is f (0)? 5. Let Ω be a region containing D (0 , 1). Let f : Ω → C be analytic. (a) Let M > 0 be a constant. Suppose | f ( z ) | ≥ M for all z ∈ ∂D (0 , 1) and | f (0) | < M . Show that f has at least one zero in D (0 , 1). (b) Let Γ be the image of ∂D (0 , 1) under f . Show that Z Γ | dz | ≥ 2 π | f (0) | . Date : November 7, 2008 (Version 1.0); due: November 14, 2008. 1...
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