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Unformatted text preview: i: (b) the semi-circular path in the upper half plane starting at 1 and ending at 3 : (c) the polygonal path from to 2 + i via the number i: 2. Evaluate R & e 2 z dz over the paths in problem 1. 3. Which of the following domains are simply connected? (a) C n [ 1 ; 1] (b) C n ( 1 ; 0] (c) f z 2 C : j z j < 1 g [ f z 2 C : j z 2 j < 1 g 4. Evaluate R C e z z 2 & 4 dz where C is (a) the positively oriented circle described by j z 3 j = 2 : (b) the positively oriented circle described by j z j = 3 : 5. Suppose f is continuous on a domain D and all of its loop integrals vanish. What can you say about f ? 6. What does the Deformation Invariance Theorem say about I j z j =1 1 sin z dz and I j z j =2 1 sin z dz ? 7. State Cauchy&s Integral Theorem. 8. Give a precise statement of Cauchy&s Integral Formula. 1...
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- Spring '11