GATE Physics Exam > GATE Physics Questions > The leading term in Laurent expansion f(z) = ...
Start Learning for Free
The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is
(d)
(a) 1 / (z ^ 3)
(b) 1 / (z ^ 2)
(c) 1/z
(d) 1?
(d)
(a) 1 / (z ^ 3)
(b) 1 / (z ^ 2)
(c) 1/z
(d) 1?
Most Upvoted Answer
The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of ar...
Understanding the Function
The function we are analyzing is f(z) = 1/(z * (1 - z)^2). To find the leading term in its Laurent expansion around z = 0, we first need to understand the components of the function.
Breaking Down the Components
- The function consists of two parts: 1/z and (1 - z)^(-2).
- The term (1 - z)^(-2) can be expanded using the binomial series.
Binomial Series Expansion
- The binomial series for (1 - z)^(-n) is given by the sum: Σ(k=0 to ∞) C(n+k-1, k) * z^k.
- For n = 2, we have: (1 - z)^(-2) = Σ(k=0 to ∞) (k+1) * z^k.
Combining the Terms
- Multiplying the expansion by 1/z gives us: f(z) = (1/z) * Σ(k=0 to ∞) (k+1) * z^k.
- This results in f(z) = Σ(k=0 to ∞) (k+1) * z^(k-1).
Identifying the Leading Term
- The first term in the series occurs when k = 0, leading to (0+1) * z^(-1) = 1/z.
- Therefore, the leading term in the Laurent expansion around z = 0 is 1/z.
Conclusion
The leading term in the Laurent expansion of f(z) around z = 0 is:
- (c) 1/z.
The function we are analyzing is f(z) = 1/(z * (1 - z)^2). To find the leading term in its Laurent expansion around z = 0, we first need to understand the components of the function.
Breaking Down the Components
- The function consists of two parts: 1/z and (1 - z)^(-2).
- The term (1 - z)^(-2) can be expanded using the binomial series.
Binomial Series Expansion
- The binomial series for (1 - z)^(-n) is given by the sum: Σ(k=0 to ∞) C(n+k-1, k) * z^k.
- For n = 2, we have: (1 - z)^(-2) = Σ(k=0 to ∞) (k+1) * z^k.
Combining the Terms
- Multiplying the expansion by 1/z gives us: f(z) = (1/z) * Σ(k=0 to ∞) (k+1) * z^k.
- This results in f(z) = Σ(k=0 to ∞) (k+1) * z^(k-1).
Identifying the Leading Term
- The first term in the series occurs when k = 0, leading to (0+1) * z^(-1) = 1/z.
- Therefore, the leading term in the Laurent expansion around z = 0 is 1/z.
Conclusion
The leading term in the Laurent expansion of f(z) around z = 0 is:
- (c) 1/z.
|
Explore Courses for GATE Physics exam
|
|
Similar GATE Physics Doubts
Top Courses for GATE PhysicsView all
The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1?
Question Description
The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1?.
The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1?.
Solutions for The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? in English & in Hindi are available as part of our courses for GATE Physics.
Download more important topics, notes, lectures and mock test series for GATE Physics Exam by signing up for free.
Here you can find the meaning of The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? defined & explained in the simplest way possible. Besides giving the explanation of
The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1?, a detailed solution for The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? has been provided alongside types of The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? theory, EduRev gives you an
ample number of questions to practice The leading term in Laurent expansion f(z) = 1/(z * (1 - z) ^ 2) of around z = 0 is(d)(a) 1 / (z ^ 3)(b) 1 / (z ^ 2)(c) 1/z(d) 1? tests, examples and also practice GATE Physics tests.
|
|
Explore Courses for GATE Physics 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