Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > The value of1/z2 dz, where the contour is the...
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The value of ∮1/z2 dz, where the contour is the unit circle traversed clockwise, is
- a)-2πi
- b)0
- c)2πi
- d)4πi
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The value of1/z2 dz, where the contour is the unit circle traversed cl...
To evaluate the integral of 1/z^2 dz, we can use the Cauchy's Integral Formula, which states that for a function f(z) that is analytic within and on a simple closed contour C, and a point a inside C,
f(a) = (1/2πi) ∮C [f(z)/(z-a)] dz.
In this case, f(z) = 1 and a = 0. The contour C is the unit circle traversed clockwise.
Applying the formula, we have:
1 = (1/2πi) ∮C [1/(z-0)] dz
= (1/2πi) ∮C 1/z dz.
The integral of 1/z with respect to z along the unit circle traversed clockwise is equal to -2πi. Therefore,
1 = (1/2πi) (-2πi)
= -1.
So, the value of 1/z^2 dz, where the contour is the unit circle traversed clockwise, is -1.
f(a) = (1/2πi) ∮C [f(z)/(z-a)] dz.
In this case, f(z) = 1 and a = 0. The contour C is the unit circle traversed clockwise.
Applying the formula, we have:
1 = (1/2πi) ∮C [1/(z-0)] dz
= (1/2πi) ∮C 1/z dz.
The integral of 1/z with respect to z along the unit circle traversed clockwise is equal to -2πi. Therefore,
1 = (1/2πi) (-2πi)
= -1.
So, the value of 1/z^2 dz, where the contour is the unit circle traversed clockwise, is -1.
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Community Answer
The value of1/z2 dz, where the contour is the unit circle traversed cl...
∮1/z2 dz
By Cauchy’s residue theorem,
∮ f(z)dz = 2πi [sum of residues]
f(z) = 1/z2
By Cauchy’s residue theorem,
∮ f(z)dz = 2πi [sum of residues]
f(z) = 1/z2
Poles are, Z = 0
Residue at Z = a, is given by
Where n is the order pole.
Here, z = 0 is second order pole.
Residue at z = 0,
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The value of1/z2 dz, where the contour is the unit circle traversed clockwise, isa)-2πib)0c)2πid)4πiCorrect answer is option 'B'. Can you explain this answer?
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