GATE EC 2025 Set 1 — Question 52
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Engineering Mathematics → Complex Analysis → Cauchy's Integral Theorem
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Question
Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle in the complex plane is/are TRUE?
Correct answer
(A) _(C) e^(z) dz = 0; (B) _(C) zⁿ dz = 0, where n is an even integer
Solution
(A) The function is analytic everywhere in the complex plane. By Cauchy's Integral Theorem, the integral of an analytic function over any closed contour is zero. Thus, . (TRUE)
(B) The integral is zero for all integers . Since is an even integer, can never be . Therefore, the integral is always zero. (TRUE)
(C) The function is analytic everywhere. By Cauchy's Integral Theorem, . (FALSE)
(D) The function has poles where , i.e., at . The nearest poles are at . Since the unit circle has radius 1, no poles lie inside . Thus, . (FALSE)
(B) The integral is zero for all integers . Since is an even integer, can never be . Therefore, the integral is always zero. (TRUE)
(C) The function is analytic everywhere. By Cauchy's Integral Theorem, . (FALSE)
(D) The function has poles where , i.e., at . The nearest poles are at . Since the unit circle has radius 1, no poles lie inside . Thus, . (FALSE)
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