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Unformatted text preview: Final exam 1. (20 Points.) Obtain all possible Laurent expansions centered at the origin for the function in part (a). Find all Laurent expansions centered at z = 1 for the function in part (b). Specify the regions of convergence of each one of these series. (a) (10 pts.) z 3 cosh 1 z ; (b) (10 pts.) z ( z 1)( z 3) : 2. (30 Points.) Evaluate the integrals (a) (15 pts.) Z d 5 + 2 cos( ) ; (b) (15 pts.) Z 1 x 1 = 2 x 2 + 1 dx: 3. (30 Points.) (a) (15 pts.) Show that, for n = 0 ; 1 ; 2 ;::: we have Z 1 1 1 (1 + x 2 ) n +1 dx = (2 n )! 2 2 n ( n !) 2 : (b) (15 pts.) Show that Z 1 sin 2 x x 2 dx = 2 : [Hint: Note that 2 sin 2 x = Re(1 e 2 ix ).] 4. (20 Points.) Find the electrostatic potential in the semidisk f x 2 + y 2 < 1 ;y > g with boundary values = 0 for f y = 0 ; 1 < x < 1 g , and = 1 for f x 2 + y 2 = 1 ;y > g . 2...
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 Fall '09
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