Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > Integration of the complex function f (z) = i...
Start Learning for Free
Integration of the complex function f (z) = 
in the counterclockwise direction, around |z – 1| = 1, is
in the counterclockwise direction, around |z – 1| = 1, is- a)-πi
- b)0
- c)πi
- d)2πi
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Integration of the complex function f (z) = in the counterclockwise di...
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
= 2πi × [sum of residues at the singualr points with in C]
Formula to find residue:
Residue Theorem:
If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then
= 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:
Given function is
2. If f(z) has a pole of order n at z = a, then
Application:
Given function is
Poles: z = 1, -1
|z – 1| = 1
⇒ |x – 1 + iy| = 1
The given region is a circle with the centre at (1, 0) and the radius is 1.
|z – 1| = 1
⇒ |x – 1 + iy| = 1
The given region is a circle with the centre at (1, 0) and the radius is 1.
Only pole z = 1, lies within the given region.
Residue at z = 1 is, 
The value of the integral = 2πi × 0.5 = πi
The value of the integral = 2πi × 0.5 = πi
Most Upvoted Answer
Integration of the complex function f (z) = in the counterclockwise di...
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
= 2πi × [sum of residues at the singualr points with in C]
Formula to find residue:
Residue Theorem:
If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then
= 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:
Given function is
2. If f(z) has a pole of order n at z = a, then
Application:
Given function is
Poles: z = 1, -1
|z – 1| = 1
⇒ |x – 1 + iy| = 1
The given region is a circle with the centre at (1, 0) and the radius is 1.
|z – 1| = 1
⇒ |x – 1 + iy| = 1
The given region is a circle with the centre at (1, 0) and the radius is 1.
Only pole z = 1, lies within the given region.
Residue at z = 1 is,
The value of the integral = 2πi × 0.5 = πi
The value of the integral = 2πi × 0.5 = πi
|
Explore Courses for Civil Engineering (CE) exam
|
|
Top Courses for Civil Engineering (CE)View all
Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer?
Question Description
Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect 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 Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect 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 Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer?.
Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect 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 Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect 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 Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer?.
Solutions for Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Civil Engineering (CE).
Download more important topics, notes, lectures and mock test series for Civil Engineering (CE) Exam by signing up for free.
Here you can find the meaning of Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Integration of the complex function f (z) = in the counterclockwise direction, around |z – 1| = 1, isa)-πib)0c)πid)2πiCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice Civil Engineering (CE) tests.
|
|
Explore Courses for Civil Engineering (CE) exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup