Let C denote the line segment from z = i to z= 1. Without evaluating the integral, show that dz < 4/2 ith rodiuus r 5.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Question 5
Let C be the upper semicircle with radius 1 centered at z = 2 given by
parametrization z(t)=2+ e" , for 0<IST, 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 -
Evaluate the contour integral of f(z)
along C for the principal branch of the power function defined by:
f(z) = z' = exp(iLog(z)), (\z| > 0, – x < Arg(z)< t ).
3.
Let C be the line segment from - 1-i to 3+ i given by the parametrization
z(t) = 2t + 1 + it , for –1<ISI, Evaluate the contour integral z dz
C
4.
Let C be the positively oriented circle centered at the origin with radius r> 3.
Without evaluating the integral, show that
22
dzs
(r2
(22
+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)
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 + i)°dz
-i
Transcribed Image Text:Let C be the upper semicircle with radius 1 centered at z = 2 given by parametrization z(t)=2+ e" , for 0<IST, 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 - Evaluate the contour integral of f(z) along C for the principal branch of the power function defined by: f(z) = z' = exp(iLog(z)), (\z| > 0, – x < Arg(z)< t ). 3. Let C be the line segment from - 1-i to 3+ i given by the parametrization z(t) = 2t + 1 + it , for –1<ISI, Evaluate the contour integral z dz C 4. Let C be the positively oriented circle centered at the origin with radius r> 3. Without evaluating the integral, show that 22 dzs (r2 (22 +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) 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 + i)°dz -i
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