math185-hw6 - Show that | P z | ≤ 1 whenever | z | ≤ 1...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 6 You may use without proof any results that had been proved in the lectures. 1. Let Ω C be a region. Let f be analytic on Ω and let z 0 Ω. Suppose f 0 ( z 0 ) 6 = 0. Show that there is an r > 0 such that Z Γ f 0 ( z 0 ) f ( z ) - f ( z 0 ) dz = 2 πi where Γ = ∂D ( z 0 , r ). 2. Let M > 0 and α > 0, α not necessarily an integer. Suppose f is entire and | f ( z ) | ≤ M | z | α for all z C . Show that f is a polynomial. 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 a C and | a | ≤ 1. Consider the polynomial P ( z ) = a 2 + (1 - | a | 2 ) z - a 2 z
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Unformatted text preview: . Show that | P ( z ) | ≤ 1 whenever | z | ≤ 1. 5. Let S = { x + iy ∈ C | x,y ∈ [0 , 1] } be the unit square in C . Let f be analytic on a region Ω that contains S . Suppose the following is true: (i) for all z with 0 ≤ Re( z ) ≤ 1, and Im( z ) = 0 or 1, Re f ( z ) = 0; (ii) for all z with Re( z ) = 0 or 1, and 0 ≤ Im( z ) ≤ 1, Im f ( z ) = 0 . Show that f ≡ 0 on Ω. Date : October 29, 2007 (Version 1.1); posted: October 25, 2007; due: October 31, 2007. 1...
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