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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e554a09-0f9c-418e-bf2f-f101c15adca6%2F431e6e86-f11b-4f8e-bac2-8341b721e2f6%2Ffhekhtf.jpeg&w=3840&q=75)
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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