380_exam_2_checklist[1] - i(b the semi-circular path in the...

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Math 380 Checklist for Exam 2 Be prepared to do the following basic kinds of calculations: parametrize directed line segments and sections of circles set up and evaluate contour integrals identify whether or not a given domain is simply connected in a given domain, determine informally whether or not two closed contours can be deformed into one another correctly apply the Fundamental Theorem for contour integrals Be prepared to state the following theorems: The Fundamental Theorem for Contour Integrals The Deformation Invariance Theorem Sample Questions 1. Evaluate R z dz where is (a) the straight line segment starting at 0 and ending at 2 +
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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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