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Unformatted text preview: Math 417, Fall 2011 Exercise Set #5 = i to z = 1. Show that 4 2. Turn in: October 5 1. Let C denote the line segment from z dz 4 C z 2. Let CR denote the upper half circle for |z| = R, for R > 1, parametrized in the counter-clockwise direction. Show that Log(z) + ln R dz 2 . 2 z R CR 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 z1 and ending at z2 . Show that: f (z) g (z) dz = f (z2 )g(z2 ) - f (z1 )g(z1 ) - g(z) f (z) dz
C C For the next three problems, all integrals can be done using results from class or the book, and without parametrizing the contour C. 4. Calculate the following integrals, where C is the positively oriented boundary of the square with vertices at 2 - 2i, 2 + 2i, -2 + 2i, -2 - 2i. e-z a) dz C z -i z2 + 8 b) dz C 2z - 1 5. Calculate the following integrals, where C is the positively oriented circle |z-i| = 2. dz a) dz 2+4 C z dz b) dz 2 2 C (z + 4) 6. Calculate the integral 0 x4 dx . x8 + 1
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This note was uploaded on 02/27/2012 for the course MATH 417 taught by Professor Staff during the Fall '11 term at SUNY Albany.
- Fall '11