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Unformatted text preview: Dr. H. Khanal Spring 2008 MA 441.03 Sample Test #2 (SOLUTION) ! Read each problem carefully before attempting a solution. Show all work to justify your answers. Answer alone carries no credit. Partial credit only for significant progress towards a correct solution. Box or underline final answers. Problems. 1 .  Find a parametric representation for the circle | z- i | = 4 (clockwise) in the complex plane. Sol. z ( t ) = i + 4 e- it , t 2 2 .  Integrate Z C f ( z ) dz , where a. f ( z ) = z 2 and C is the upper half of the unit circle | z | = 1 b. f ( z ) = e 3 z and C is the shortest path from 0 to i 3 Sol. a. C : z ( t ) = e it , t , dz = ie it dt , f ( z ) = z 2 = ( e it ) 2 = ( e- it ) 2 = e- 2 it , thus Z C f ( z ) dz = Z e- 2 it ie it dt = Z ie- it dt =- e- it =- (cos - i sin ) + 1 = 2 b. Since f ( z ) = e 3 z is analytic, we can integrate directly as Z C f ( z ) dz = Z i 3 e 3 z dz = 1 3 e z i 3 = 1 3 ( e i- 1) = 1 3 (cos...
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This note was uploaded on 06/03/2008 for the course MA 441 taught by Professor Kaba during the Fall '08 term at Embry-Riddle FL/AZ.
- Fall '08