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8. Let C be the perimeter of the square with vertices at the points z = z = 1+i, and z = i traversed once in that order. Show that = 0, z = 1, %3D %3D e dz = 0. alumal

8. Let C be the perimeter of the square with vertices at the points z = z = 1+i, and z = i traversed once in that order. Show that = 0, z = 1, %3D %3D e dz = 0. alumal

Intermediate Algebra
Intermediate Algebra
10th Edition
ISBN:9781285195728
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter8: Conic Sections
Section8.2: More Parabolas And Some Circles
Problem 63.1PS
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Question 8

4.2 Contour Integrals (1+i) cos(it) dt (a) +2+ -2 (12 +i)² 4. Furnish the details of the proof of Theorem 3. 5. Utilize Example 2 to evaluate 9. 2. +1- 3(z - i)² dz, 7(? – 2) where C is the circle z – i = 4 traversed once counterclockwise. - 6. Compute fzdz, where (a) T is the circle z] = 2 traversed once counterclockwise. (b) T is the circle z| = 2 traversed once clockwise. (c) T is the circle z = 2 traversed three times clockwise. %3D 7. Compute f, Re z dz along the directed line segment from z = 0 to z =1+2i. 8. Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1+i, and z = i traversed once in that order. Show that %3D e dz = 0. /9. Evaluate (x- 2xyi) dz over the contour T: z = t + it2, 0 <t < 1, where X3 Re z, y = Im z. 10. Compute fcZ dz along the perimeter of the square in Prob. 8. 11. Evaluate -(2z + 1) dz, where F is the following contour from z = -i to z = 1: (a) the simple line segment, (b) two simple line segments, the first from z = -i to z = 0 and the second from z =0 to z = 1.
Transcribed Image Text:4.2 Contour Integrals (1+i) cos(it) dt (a) +2+ -2 (12 +i)² 4. Furnish the details of the proof of Theorem 3. 5. Utilize Example 2 to evaluate 9. 2. +1- 3(z - i)² dz, 7(? – 2) where C is the circle z – i = 4 traversed once counterclockwise. - 6. Compute fzdz, where (a) T is the circle z] = 2 traversed once counterclockwise. (b) T is the circle z| = 2 traversed once clockwise. (c) T is the circle z = 2 traversed three times clockwise. %3D 7. Compute f, Re z dz along the directed line segment from z = 0 to z =1+2i. 8. Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1+i, and z = i traversed once in that order. Show that %3D e dz = 0. /9. Evaluate (x- 2xyi) dz over the contour T: z = t + it2, 0 <t < 1, where X3 Re z, y = Im z. 10. Compute fcZ dz along the perimeter of the square in Prob. 8. 11. Evaluate -(2z + 1) dz, where F is the following contour from z = -i to z = 1: (a) the simple line segment, (b) two simple line segments, the first from z = -i to z = 0 and the second from z =0 to z = 1.
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