math185-hw9sol

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

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 9 SOLUTIONS For a ∈ C , r > 0, we write D * ( a,r ) := { z ∈ C | < | z- a | < r } . We write C × = C \{ } . 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....
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This note was uploaded on 08/01/2008 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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

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