Question 3

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 3
parametrization z(t) = 2 + e" , for 0<tST, Evaluate the contour integral
dz
2.
Let C be the right-hand semicircle with radius 1 centered at origin given by the
parametrization z(t) = e" , for - sts. Evaluate the contour integral of ſ(z)
along C for the principal branch of the power function defined by:
f(z) = z' = exp(iLog(z)), (z|> 0, – n < Arg(z)< t ).
%3D
3.
Let C be the line segment from - 1-i to 3 + i given by the parametrization
z(t) = 2t +1+ it , for –1<t<1, Evaluate the contour integral J 2 az
4.
Let C be the positively oriented circle centered at the origin with radius r> 3.
Without evaluating the integral, show that
277-3
dz<
(r² –9)2
5.
Let C denote the line segment from z = i to z = 1. Without evaluating the integral,
show that
<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)
TT + In(r)
dz < 2n
By finding an antiderivative, evaluate the integral, where the contour is any path
between the indicated limits of integration:
7.
i
(2: + 1)°dz
-i
Transcribed Image Text:parametrization z(t) = 2 + e" , for 0<tST, Evaluate the contour integral dz 2. Let C be the right-hand semicircle with radius 1 centered at origin given by the parametrization z(t) = e" , for - sts. Evaluate the contour integral of ſ(z) along C for the principal branch of the power function defined by: f(z) = z' = exp(iLog(z)), (z|> 0, – n < Arg(z)< t ). %3D 3. Let C be the line segment from - 1-i to 3 + i given by the parametrization z(t) = 2t +1+ it , for –1<t<1, Evaluate the contour integral J 2 az 4. Let C be the positively oriented circle centered at the origin with radius r> 3. Without evaluating the integral, show that 277-3 dz< (r² –9)2 5. Let C denote the line segment from z = i to z = 1. Without evaluating the integral, show that <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) TT + In(r) dz < 2n By finding an antiderivative, evaluate the integral, where the contour is any path between the indicated limits of integration: 7. i (2: + 1)°dz -i
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