By finding an antiderivative, evaluate the integral, where the contour is any path between the indicated limits of integration: i [(2z +i)° dz -i 7.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Question 7
t. Evaluate the contour inte
dz
2.
Let C be the right-hand semicircle with radius 1 centered at origin given by the
parametrization z(t) = e"
', for-sts을
Evaluate the contour integral of f(z)
2
along C for the principal branch of the power function defined by:
f(z) = z' = exp(iLog(z)), (|z| > 0, – x < Arg(z)<n ).
3.
Let C be the line segment from - 1-i to 3+ i given by the parametrization
z(t) = 2t +1+ it , for –1<i<1. Evaluate the contour integral Zz dz ,
4.
Let C be the positively oriented circle centered at the origin with radiusr>3.
Without evaluating the integral, show that
22
2o3
dz<
(z-
+9)-
(72 -9)2
5.
Let C denote the line segment from z = i to z = 1. Without evaluating the integral,
show that
dz < 4/2
Let C be the positively oriented circle centered at the origin with radius r > 1.
Without evaluating the integral, show that
6.
Log(z)
T + In(r)
dz < 27
By finding an antiderivative, evaluate the integral, where the contour is any path
between the indicated limits of integration:
7.
i
(2z + i)°dz
Transcribed Image Text:t. Evaluate the contour inte dz 2. Let C be the right-hand semicircle with radius 1 centered at origin given by the parametrization z(t) = e" ', for-sts을 Evaluate the contour integral of f(z) 2 along C for the principal branch of the power function defined by: f(z) = z' = exp(iLog(z)), (|z| > 0, – x < Arg(z)<n ). 3. Let C be the line segment from - 1-i to 3+ i given by the parametrization z(t) = 2t +1+ it , for –1<i<1. Evaluate the contour integral Zz dz , 4. Let C be the positively oriented circle centered at the origin with radiusr>3. Without evaluating the integral, show that 22 2o3 dz< (z- +9)- (72 -9)2 5. Let C denote the line segment from z = i to z = 1. Without evaluating the integral, show that dz < 4/2 Let C be the positively oriented circle centered at the origin with radius r > 1. Without evaluating the integral, show that 6. Log(z) T + In(r) dz < 27 By finding an antiderivative, evaluate the integral, where the contour is any path between the indicated limits of integration: 7. i (2z + i)°dz
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