GATE EC 2017 Set 2 — Question 27
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Engineering Mathematics → Complex Analysis → Residue Theorem & Contour Integration
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Question
An integral over a counterclockwise circle is given byIf is defined as , then the value of is
Correct answer
(D) -4π i sin(1)
Solution
The given integral is , where .
The contour is a counterclockwise circle .First, find the singularities of by setting the denominator to zero:
.Both singularities and lie inside the contour since and .
These are simple poles.We use Cauchy's Residue Theorem, which states , where are the poles inside .Residue at :Residue at :Now, sum the residues:Recall Euler's formula for : .
So, .Substitute this back into the sum of residues:Finally, apply Cauchy's Residue Theorem:The final answer is .
The contour is a counterclockwise circle .First, find the singularities of by setting the denominator to zero:
.Both singularities and lie inside the contour since and .
These are simple poles.We use Cauchy's Residue Theorem, which states , where are the poles inside .Residue at :Residue at :Now, sum the residues:Recall Euler's formula for : .
So, .Substitute this back into the sum of residues:Finally, apply Cauchy's Residue Theorem:The final answer is .
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