GATE EC 2017 Set 2 — Question 4
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Engineering Mathematics → Complex Analysis → Residue Theorem & Contour Integration
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Question
The residues of a function are
Correct answer
(B) (1)/(125) and (-1)/(125)
Solution
The given function is .
This function has two poles:
For a simple pole , the residue is given by .
.Residue at the pole of order 3 at :
For a pole of order at , the residue is given by .
Here, and .
Let's find the derivatives:
First derivative: .
Second derivative: .Now, substitute into the second derivative:
.Thus, the residues of the function are and .Comparing with the given options:
(A) and
(B) and
(C) and
(D) and The correct option is (B).The final answer is
This function has two poles:
1.A simple pole at .
2.A pole of order 3 at .
Residue at the simple pole :For a simple pole , the residue is given by .
.Residue at the pole of order 3 at :
For a pole of order at , the residue is given by .
Here, and .
Let's find the derivatives:
First derivative: .
Second derivative: .Now, substitute into the second derivative:
.Thus, the residues of the function are and .Comparing with the given options:
(A) and
(B) and
(C) and
(D) and The correct option is (B).The final answer is
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