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math185-hw9sol - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 9 SOLUTIONS For a C , r > 0, we write D * ( a, r ) := { z C | 0 < | z - a | < r } . We write C × = C \{ 0 } . You may use without proof any results that had been proved in the lectures. 1. 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. Solution. f a is analytic in C × and its Laurent expansion about z = 0 may be obtained as follows: e e 1 z = 1 + e 1 z + 1 2! e 2 z + · · · + 1 n ! e n z + · · · = 1 + 1 + 1 z + 1 2! 1 z 2 + · · · + 1 2! " 1 + 2 z + 1 2! 2 z 2 + · · · # + · · · + 1 n ! 1 + n z + 1 2! n z 2 + · · · + · · · . Observe that the coefficient of the term z - 1 is simply 0 + 1 + 2 2! + 3 3! + · · · + n n ! + · · · = 1 + 1 1! + 1 2! + · · · + 1 ( n - 1)! + · · · = e. By the residue theorem Z Γ a f a = 2 πi Res( f a ; 0) Ind(Γ a ; 0) = 2 πei. (b) f b : D * (0 , π ) C is given by f b ( z ) = 1 (sin z ) 3 and Γ b is the boundary ∂D (0 , 1) traversed once counter-clockwise.
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