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Unformatted text preview: Exam 1: MAP 4403 * November 5, 2004 Name : Student ID : Note: Calculators are allowed but not needed. This is a closed book exam: Formula sheets are not allowed. 1. Find Z C Re( z 2 ) dz, where C is the horizontal line segment from the point i to the point 1 + i . Put z ( t ) = t + i with t ∈ [0 , 1]. Then Re( z 2 ) = t 2- 1 and dz = dt , so Z C Re( z 2 ) dz = Z 1 t 2- 1 dt = t 3 / 3- t | 1 =- 2 / 3 . 2. Find and draw all cubic roots of 8e i π 2 . 2e i ( π 6 + k 2 π 3 ) , k = 0 , 1 , 2. These are on the circle of radius 2 with center 0. The root corre- sponding to k = 0 has argument equal to π/ 6 and the second and third are found by rotating the first over an angle of +2 π/ 3, respectively +4 π/ 3. 3. Find the principal value of Z + ∞-∞ dx ( x- 1)( x + 2)( x 2- 2 x + 2) . (Hint: Recall the formula pr. v. Z + ∞-∞ f ( x ) dx = 2 πi X Res f ( z ) + πi X Res f ( z ) where the first sum of residues is taken over singularities of f in the upper half plane and the second over simple poles of...
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This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.
- Summer '06