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Which of the following is not true?
- a)log(1 + z) = z −
…..about z = 0 - b)
+……about z = 2 - c)
=0 where C is the circle |z-1| = 2 - d)
has no singularity
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Which of the following is not true?a)log(1 + z) = z −…..a...
Concept:
Taylor series expansion
Option 1:
The standard expansion of log(1 + z) is given as
Hence, Option 1 is true
Option 2:
Given complex function is
→ Let’s Resolve f(z) into partial fractions
For expanding about z = 2, let z – 2 = t ⇒ z = 2 + t
Option 3:
Cauchy’s Integral Formula:
Option 1:
The standard expansion of log(1 + z) is given as
Hence, Option 1 is true
Option 2:
Given complex function is
→ Let’s Resolve f(z) into partial fractions
For expanding about z = 2, let z – 2 = t ⇒ z = 2 + t
Option 3:
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:
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
Given complex integral is
where Cis the circle |z-1| = 2;
2. If f(z) has a pole of order n at z = a, then
Given complex integral is
where Cis the circle |z-1| = 2;Now for the given complex function, the pole is -4 with order 2;
The pole - 4 lies outside the given circle C;
Therefore, no residue inside the circle, hence integration will be zero.
Option 3 is also correct
Option 4:
The given complex function is f(z) = 
In this function, the singularities are z = 0, +i, -i;
In this function, the singularities are z = 0, +i, -i;
Therefore, the given function has 3 singularities...
Option 4 is incorrect
Most Upvoted Answer
Which of the following is not true?a)log(1 + z) = z −…..a...
Concept:
Taylor series expansion
Option 1:
The standard expansion of log(1 + z) is given as
Hence, Option 1 is true
Option 2:
Given complex function is
→ Let’s Resolve f(z) into partial fractions
For expanding about z = 2, let z – 2 = t ⇒ z = 2 + t
Option 3:
Cauchy’s Integral Formula:
Option 1:
The standard expansion of log(1 + z) is given as
Hence, Option 1 is true
Option 2:
Given complex function is
→ Let’s Resolve f(z) into partial fractions
For expanding about z = 2, let z – 2 = t ⇒ z = 2 + t
Option 3:
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:
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
Given complex integral is
where Cis the circle |z-1| = 2;
2. If f(z) has a pole of order n at z = a, then
Given complex integral is
where Cis the circle |z-1| = 2;Now for the given complex function, the pole is -4 with order 2;
The pole - 4 lies outside the given circle C;
Therefore, no residue inside the circle, hence integration will be zero.
Option 3 is also correct
Option 4:
The given complex function is f(z) =
In this function, the singularities are z = 0, +i, -i;
In this function, the singularities are z = 0, +i, -i;
Therefore, the given function has 3 singularities...
Option 4 is incorrect
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Which of the following is not true?a)log(1 + z) = z −…..about z = 0b)+……about z = 2c)=0 where C is the circle |z-1| = 2d)has no singularityCorrect answer is option 'D'. Can you explain this answer?
Question Description
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Which of the following is not true?a)log(1 + z) = z −…..about z = 0b)+……about z = 2c)=0 where C is the circle |z-1| = 2d)has no singularityCorrect answer is option 'D'. 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 Which of the following is not true?a)log(1 + z) = z −…..about z = 0b)+……about z = 2c)=0 where C is the circle |z-1| = 2d)has no singularityCorrect answer is option 'D'. 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 Which of the following is not true?a)log(1 + z) = z −…..about z = 0b)+……about z = 2c)=0 where C is the circle |z-1| = 2d)has no singularityCorrect answer is option 'D'. Can you explain this answer?.
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