math185f09-hw7sol

math185f09-hw7sol - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 7 SOLUTIONS 1. Let β ∈ C . (a) Show that for all n = 0 , 1 , 2 ,..., β n n ! 2 = 1 2 πi Z ∂D (0 , 1) β n e βz n ! z n +1 dz. Solution. Applying generalized Cauchy’s integral formula, we get 1 2 πi Z ∂D (0 , 1) β n e βz n ! z n +1 dz = β n n ! " 1 2 πi Z ∂D (0 , 1) e βz z n +1 dz # = β n n ! × 1 n ! d n dz n e βz z =0 = β n n ! 2 . (b) Show that ∞ X n =0 β n n ! 2 = 1 2 π Z 2 π e 2 β cos θ dθ. [ Hint : Consider power series expansion of e β/z and apply (a) on z- 1 e β ( z +1 /z ) .] Solution. Note that e β/z = ∞ X n =0 β n n ! z n . Multiplying by e βz and dividing by z , we get 1 z e β ( z +1 /z ) = ∞ X n =0 β n e βz n ! z n +1 . Integrating about ∂D (0 , 1) and using (a), we get 1 2 πi Z ∂D (0 , 1) 1 z e β ( z +1 /z ) dz = ∞ X n =0 β n n ! " 1 2 πi Z ∂D (0 , 1) e βz z n +1 dz # = ∞ X n =0 β n n ! 2 Evaluating the line integral about the path z : [0 , 2 π ] → C , z ( θ ) = e iθ and noting that e iθ + e- iθ = 2cos θ we get 1 2 πi Z ∂D (0 , 1) 1 z e β ( z +1 /z ) dz = 1 2 πi Z 2 π e 2 β cos θ dθ. 2. Let f : C → C be an entire function. Let a ∈ R be an arbitrary constant. (a) Show that if Re f ( z ) ≤ a for all z ∈ C , then f is constant. (b) Show that if Re f ( z ) ≥ a for all z ∈ C , then f is constant. (c) Show that if [Re f ( z )] 2 ≤ [Im f ( z )] 2 for all z ∈ C , then f is constant. (d) Show that if [Re f ( z )] 2 ≥ [Im f ( z )] 2 for all z ∈ C , then f is constant. (e) Suppose h is another entire functions and suppose there exists an a ∈ R , a > 0, such that Re f ( z ) ≤ a Re h ( z ) for all z ∈ C . Show that there exist α,β ∈ C such that f ( z ) = αh ( z ) + β for all z ∈ C . Date : November 14, 2009 (Version 1.0). 1 [ Hint : if f and g are both entire, then so are f ◦ g and g ◦ f ; find an appropriate g so that you may apply Liouville’s theorem.] Solution. Note that e x is a monotone increasing function on R ....
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This note was uploaded on 02/17/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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math185f09-hw7sol - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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