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homework20111005

# homework20111005 - Math 417 Fall 2011 Exercise Set#5 = i to...

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Math 417, Fall 2011 Exercise Set #5 Turn in: October 5 1. Let C denote the line segment from z = i to z = 1. Show that C dz z 4 4 2 . 2. Let C R denote the upper half circle for | z | = R , for R > 1, parametrized in the counter-clockwise direction. Show that C R Log ( z ) z 2 dz 2 π π + ln R R . Conclude that the integral tends to zero as R tends to infinity. 3. Suppose that f ( z ) and g ( z ) are analytic functions defined in a domain , and that C is a contour in starting at z 1 and ending at z 2 . Show that: C f ( z ) g ( z ) dz = f ( z 2 ) g ( z 2 ) f ( z 1 ) g ( z 1 ) C g ( z ) f ( z ) dz For the next three problems, all integrals can be done using results from class or the
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