GATE ME 2021 Set 1 — Question 37
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Engineering Mathematics → Complex Variables → Residue Theorem
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Question
Let represent the unit circle centered at origin in the complex plane, and complex variable, . The value of the contour integral (where integration is taken counter clockwise) is
Correct answer
(C) π i
Solution
We use Cauchy's Integral Formula: , where is analytic within and on the closed contour and lies inside .
Here, the integral is .
Let and . The function is an entire function (analytic everywhere). The point lies inside the unit circle .
Applying the formula:Therefore, the original integral is:
Here, the integral is .
Let and . The function is an entire function (analytic everywhere). The point lies inside the unit circle .
Applying the formula:Therefore, the original integral is:
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