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GATE 2025 MA – Question 33

Complex Analysis · Complex Integration and Residues · 1 mark · Numerical answer

$I=(2\pi i)^{-1}\oint_C\sin z/[1-\cos(z^3)]\,dz$, where C is the positively oriented diamond |Re z|+|Im z|=1. Find 120I (integer).

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Correct answer: 2 to 2

Explanation

Only z=0 is singular inside: other zeros satisfy z³=2πk and have modulus at least (2π)^(1/3)>1. Near zero, $[1-\cos(z^3)]^{-1}=2z^{-6}+O(1)$. Multiplying by sinz=z−z³/6+z⁵/120−… gives residue 2/120=1/60. Hence I=1/60 and **120I=2**.

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