GATE EC 2018 Set 1 — Question 51
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Engineering Mathematics → Complex Analysis → Residue Theorem & Contour Integration
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Question
The contour given below is on the complex plane , where .
The value of the integral is __________.
The value of the integral is __________.
Correct answer
14 to 17
Solution
The given integral is .First, identify the integrand . We can factor the denominator as .
So, the poles of are at and . Both are simple poles.Next, calculate the residues at each pole:
Applying this to our contour :
Finally, substitute this result back into the original expression for :
The value of the integral is .The final answer is .
So, the poles of are at and . Both are simple poles.Next, calculate the residues at each pole:
1.Residue at :
.2.Residue at :
.Now, analyze the contour from the given figure. The contour is an "infinity" symbol that self-intersects at the origin. It can be decomposed into two simple closed contours:- : The left loop, which encircles the pole in the counter-clockwise (CCW) direction. The winding number for is .
- : The right loop, which encircles the pole in the clockwise (CW) direction. The winding number for is .
Applying this to our contour :
Finally, substitute this result back into the original expression for :
The value of the integral is .The final answer is .
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