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241F09Final

241F09Final - Final Math 241 Fall 2009 Instructors Block...

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Final, Math 241, Fall 2009 Instructors - Block, Lieberman You may use one sheet of 8 x 11” paper on which you write any infor- mation you like. No calculator.Good luck. Show all work , even on multiple choice questions. (1) Compute the principal value of the integral Z 0 sin x x ( x 2 + 1) dx. (a) 0 (b) 1 2 (2 - e - 1 ) (c) π 2 e (d) π 2 (1 - e - 1 ) (e) π 2 (2 - e - 1 ) 1

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2 (2) Evaluate R C sin(2 z ) (6 z - π ) 3 dz , where C is the ellipse given by x 2 + 4 y 2 = 4 oriented counter-clockwise. (a) 0 (b) 1 / 2 (c) πi (d) - 3 (e) - 2 πi 3
3 (3) Evaluate the integral of f ( z ) = z cos( z 2 ) along the contour C that begins at 0, moves along the real axis to 1, moves counter- clockwise around the circle of radius 1 until it reaches - 1, then moves down along a vertical path to - 1 - i . (Hint: there is a shortcut.) (a) 0 (b) i 2 ( e - 2 - e 2 ) (c) 1 2 (1 + i )( e 2 - e - 2 ) (d) i 4 ( e 2 - e - 2 ) (e) 1 2 ( e 2 - e - 2 )

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4 (4) Compute a Laurent expansion of the function f ( z ) = 1 ( z - 2 i )( z + i ) valid on the annulus given by 1 < | z | < 2.
5 (5) (a) Compute all possible values of i πi 2 . (b) Compute all possible solutions of the equation cos(

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241F09Final - Final Math 241 Fall 2009 Instructors Block...

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