GATE EC 2023 Set 1 — Question 14
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Engineering Mathematics → Complex Analysis → Residue Theorem & Contour Integration
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Question
The value of the contour integral, , where the contour is , taken in the counter clockwise direction, is
Correct answer
(B) π(1 + j)
Solution
Let .
First, find the poles of by setting the denominator to zero:
Using the quadratic formula :
So, the poles are and .Next, determine which poles lie inside the contour .
The contour is a circle defined by . This is a circle centered at with radius .Let's check :
. Since , the pole is inside the contour .Let's check :
. Since , the pole is outside the contour .So, only is inside the contour.
We can rewrite the denominator as .
The integral is .
By Cauchy's Residue Theorem, the integral is .
The residue at is:
Substitute and :
(since )
Now, calculate the integral:
Integral
The final answer is .
First, find the poles of by setting the denominator to zero:
Using the quadratic formula :
So, the poles are and .Next, determine which poles lie inside the contour .
The contour is a circle defined by . This is a circle centered at with radius .Let's check :
. Since , the pole is inside the contour .Let's check :
. Since , the pole is outside the contour .So, only is inside the contour.
We can rewrite the denominator as .
The integral is .
By Cauchy's Residue Theorem, the integral is .
The residue at is:
Substitute and :
(since )
Now, calculate the integral:
Integral
The final answer is .
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