GATE 2025 EC – Question 52

Engineering Mathematics · Complex Analysis · 2 marks · Multiple select

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

$\oint_Ce^z\,dz=0$
$\oint_Cz^n\,dz=0$, where $n$ is an even integer
$\oint_C\cos z\,dz\ne0$
$\oint_C\sec z\,dz\ne0$

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Correct answer: (A) $\oint_Ce^z\,dz=0$; (B) $\oint_Cz^n\,dz=0$, where $n$ is an even integer

Explanation

(A) $e^z$ is entire, so by Cauchy's theorem the integral is 0: true. (B) $\oint_Cz^n\,dz=2\pi j$ only for $n=-1$ and 0 for every other integer $n$; an even $n$ is never $-1$, so it is 0: true. (C) $\cos z$ is entire, so the integral is 0, and "$\ne0$" is false. (D) The poles of $\sec z$ are at $\pm\pi/2,\dots$, all outside the unit circle, so $\sec z$ is analytic inside $C$ and the integral is 0: "$\ne0$" is false.