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math185f09-hw10

math185f09-hw10 - b is the boundary ∂D(0 1 traversed once...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 10 1. (a) Show that if f has a pole or an essential singularity at a , then e f has an essential singularity at a . (b) Let Ω C be a region. Let a Ω and f : Ω \{ a } → C be a function with an isolated singularity at a . Suppose for some m N and ε > 0, Re f ( z ) ≤ - m log | z - a | for all z D * ( a, ε ). Show that a is a removable singularity of f . 2. (a) Let f : D * (0 , 1) C be analytic. Show that if | f ( z ) | ≤ log 1 | z | for all z D * (0 , 1), then f 0. (b) Let f : C × C be analytic on C × with a pole of order 1 at 0. Show that if f ( z ) R for all | z | = 1, then for some α C × and β R , f ( z ) = αz + α 1 z + β for all z C × . 3. Evaluate the integral Z Γ i f i for i = a, b . (a) f a : C × C is given by f a ( z ) = e e 1 z and Γ a is the boundary ∂D (0 , 2) traversed once counter-clockwise. (b) f b : D * (0 , π ) C is given by f b ( z ) = 1 (sin z ) 3 and Γ

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Unformatted text preview: b is the boundary ∂D (0 , 1) traversed once counter-clockwise. 4. (a) Let the Laurent expansion of cot( πz ) on A (0;1 , 2) be cot( πz ) = ∞ X n =-∞ a n z n . Compute a n for n < 0. (b) For n = 0 , 1 , 2 ,..., compute 1 2 πi Z Γ n dz z 3 sin z where Γ n is the circle ∂D (0 ,r n ) traversed once counter-clockwise and r n = ( n + 1 2 ) π . Date : Posted: December 5, 2009. 1 5. (a) Does the following function have an antiderivative on A (0;4 , ∞ )? z ( z-1)( z-2)( z-3) (b) Does the following function have an antiderivative on A (0;4 , ∞ )? z 2 ( z-1)( z-2)( z-3) 2...
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