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Unformatted text preview: z = 2 e it , 0 t (b) the semicircle z = 2 e it , t 2 (c) the semicircle z = 2 e it , 0 t 2 7. Let f ( z ) = e z and evaluate R C f ( z ) dz , where C is the boundary of the square with vertices at the points 0, 1, 1 + i , and i , the orientation being in the clockwise direction. 8. Prove that if f is the constant function f ( z ) = 1, and if C is any contour from z 1 to z 2 , then R C f ( z ) dz = z 2-z 1 . 9. Let C R be the circle | z | = R (for R > 0) oriented in the counterclockwise direction. Prove that lim R Z C R Log z z 2 dz = 0 . Hint: First show that R C R Log z z 2 dz 2 ( +ln R R ) ....
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This note was uploaded on 02/21/2012 for the course MATH 118 taught by Professor Forde during the Fall '09 term at University of Houston.
- Fall '09