GATE EC 2014 Set 1 — Question 13
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Engineering Mathematics → Complex Analysis → Cauchy's Integral Formula
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Question
is a closed path in the -plane given by . The value of the integral is
Correct answer
(C) -4π(3 + j2)
Solution
The integral is of the form , where and .
In the notation used in the options ( instead of ):
Thus, the correct option is (C).
1.Identify the singularity: The pole is at .
2.Check if the pole is inside the contour: The contour is a circle defined by . The magnitude of the pole is . Since , the pole lies inside the contour .
3.Apply Cauchy's Integral Formula:
Substitute into :4.Calculate the final value:
Since :In the notation used in the options ( instead of ):
Thus, the correct option is (C).
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