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GATE 2021 MA – Question 56

Complex Analysis · Complex Integration and Residues · 2 marks · Numerical answer

R={z=x+iy:0<x<1,−11π<y<11π}, with positively oriented boundary Γ. Evaluate $(2\pi i)^{-1}\oint_\Gamma e^z/(e^z-2)\,dz$.

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Correct answer: 11 to 11

Explanation

Poles occur at z=ln2+2πik,k∈ℤ, each simple with residue 1. The real part lies between 0 and 1, while the imaginary bound requires −5.5<k<5.5, giving k=−5,…,5. Eleven poles are enclosed, so the normalized integral is **11**.

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