GATE EC 2015 Set 3 — Question 13
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Engineering Mathematics → Complex Analysis → Cauchy's Integral Formula
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Question
If is a circle of radius with centre , in the complex -plane and if is a non-zero integer, then equals
Correct answer
(B) 0
Solution
According to Cauchy's Integral Formula, if a function is analytic within and on a simple closed contour , then . In this problem, , which is an entire function (analytic everywhere). For any non-zero integer , the -th derivative of a constant is zero (). If is a negative integer, say where , then the integrand is , which is analytic everywhere, and the integral is zero by Cauchy's Theorem. Thus, for any non-zero integer , the integral is 0.
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