GATE 2020 MA – Question 55

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

Let $f(z)=\prod_{n=1}^{17}(z-\pi/n)$ and $\gamma(t)=e^{2it}$ for 0 ≤ t ≤ 2π. If $\int_\gamma f\prime(z)/f(z)\,dz=\alpha\pi i$, find α.

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Correct answer: 56 to 56

Explanation

Zeros satisfy π/n < 1 precisely for n ≥ 4. Thus 14 zeros, n = 4,…,17, lie inside the unit circle; none lies on it. The curve $e^{2it}$ traverses that circle twice counterclockwise. The argument principle gives integral $2\pi i(2)(14)=56\pi i$, so α = 56.

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