GATE EC 2019 Set 1 — Question 26
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Engineering Mathematics → Complex Analysis → Residue Theorem & Contour Integration
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Question
The value of the contour integral evaluated over the unit circle is ________.
Correct answer
0.0001 to 0.0001
Solution
The given integral is , where is the unit circle .
First, expand the integrand:
So the integral becomes:
According to Cauchy's Residue Theorem, , where are the poles of inside the contour .
The function has a pole at . This is a pole of order 2.
To find the residue at , we can use the formula for a pole of order :
Here, and .
Alternatively, by inspecting the Laurent series , the coefficient of is , which is the residue.
Since the residue at the only pole inside the unit circle is , the integral .
Therefore, .The final answer is
First, expand the integrand:
So the integral becomes:
According to Cauchy's Residue Theorem, , where are the poles of inside the contour .
The function has a pole at . This is a pole of order 2.
To find the residue at , we can use the formula for a pole of order :
Here, and .
Alternatively, by inspecting the Laurent series , the coefficient of is , which is the residue.
Since the residue at the only pole inside the unit circle is , the integral .
Therefore, .The final answer is
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