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The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is: 
- a)8πi
- b)-8πi
- c)-πi
- d)πi
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The value of the following complex integral, with C representing the u...
Concept:
Cauchy’s Theorem:
If f(z) is an analytic function and f’(z) is continuous at each point within and on a closed curve C, then
Cauchy’s Integral Formula:
If f(z) is an analytic function within a closed curve and if a is any point within C, then
Residue Theorem:
If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then
f(z)dz = 2πi × [sum of residues at the singualr points with in C]
Cauchy’s Integral Formula:
If f(z) is an analytic function within a closed curve and if a is any point within C, then
Residue Theorem:
If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then
f(z)dz = 2πi × [sum of residues at the singualr points with in C]Formula to find residue:
1. If f(z) has a simple pole at z = a, then
2. If f(z) has a pole of order n at z = a, then
Application:
The simple poles are: z = 0, 2
The given region is a unit circle.
The residue at z = 2 is zero as it lies outside the given region.
The reside at z = 0, is given by
The value of the given integral =
The value of the given integral =
Most Upvoted Answer
The value of the following complex integral, with C representing the u...
Concept:
Cauchy’s Theorem:
If f(z) is an analytic function and f’(z) is continuous at each point within and on a closed curve C, then
Cauchy’s Integral Formula:
If f(z) is an analytic function within a closed curve and if a is any point within C, then
Residue Theorem:
If f(z) is analytic in a closed curve C except at a finite number of singular points within C, thenf(z)dz = 2πi × [sum of residues at the singualr points with in C]
Cauchy’s Integral Formula:
If f(z) is an analytic function within a closed curve and if a is any point within C, then
Residue Theorem:
If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then
f(z)dz = 2πi × [sum of residues at the singualr points with in C]Formula to find residue:
1. If f(z) has a simple pole at z = a, then
2. If f(z) has a pole of order n at z = a, thenApplication:
The simple poles are: z = 0, 2
The given region is a unit circle.
The residue at z = 2 is zero as it lies outside the given region.
The reside at z = 0, is given by
The value of the given integral =
The value of the given integral =
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The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:a)8πib)-8πic)-πid)πiCorrect answer is option 'C'. Can you explain this answer?
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The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:a)8πib)-8πic)-πid)πiCorrect answer is option 'C'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:a)8πib)-8πic)-πid)πiCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:a)8πib)-8πic)-πid)πiCorrect answer is option 'C'. Can you explain this answer?.
The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:a)8πib)-8πic)-πid)πiCorrect answer is option 'C'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:a)8πib)-8πic)-πid)πiCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:a)8πib)-8πic)-πid)πiCorrect answer is option 'C'. Can you explain this answer?.
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