hw9_1 - a cos θ = 2 π √ a 2-1 Hint You can realize this...

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Complex Variables Assignment 9.1 Spring 2011 Due October 31 Exercise 1. Use Cauchy’s Integral Formula to evaluate the following definite integrals. All contours are positively oriented. a. Z γ sin e z z dz where γ is any simple closed curve containing the origin. b. Z γ z ( z - a )( z - b ) dz where γ is any simple closed curve containing the distinct complex numbers a and b . c. Z γ e z 2 z 3 - 1 dz where γ is the circle | z - 1 | = 1. Exercise 2. Show that if a > 1 then Z 2 π 0
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Unformatted text preview: a + cos θ = 2 π √ a 2-1 . [ Hint: You can realize this integral as the parametrized form of an integral over the unit circle by setting z = e iθ .] Exercise 3. Suppose that f is analytic on a disk D and that γ is a simple closed curve in D . Suppose that f is constant on γ . Prove that f takes on the same constant value everywhere inside γ as well....
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