GATE EE 2019 Set 1 — Question 37
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Engineering Mathematics → Complex Variables → Cauchy's Integral Theorem
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Question
The closed loop line integralevaluated counter-clockwise, is
Correct answer
(A) +8jπ
Solution
To evaluate the closed loop line integral , we identify the singularities of the integrand .The singularity occurs at , which gives .The contour is a circle centered at the origin with radius 5. Since the magnitude of the singularity is less than the radius 5, the singularity lies inside the contour.According to Cauchy's Integral Formula:where and .Evaluating at :
.Therefore, the integral is:
.Thus, the correct option is (A).
.Therefore, the integral is:
.Thus, the correct option is (A).
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