GATE 2022 ME (ME1) – Question 36

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Multiple choice

The value of the integral $\oint\left(\frac{6z}{2z^4 - 3z^3 + 7z^2 - 3z + 5}\right)dz$ evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole $z = i$, where $i$ is the imaginary unit, is

$(-1 + i)\pi$
$(1 + i)\pi$
$2(1 - i)\pi$
$(2 + i)\pi$

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Correct answer: (A) $(-1 + i)\pi$

Explanation

Since $z = i$ is a pole, the denominator factorises as $(z^2 + 1)(2z^2 - 3z + 5)$. The residue at $z = i$ is $\frac{6i}{2i \cdot (2(-1) - 3i + 5)} = \frac{3}{3 - 3i} = \frac{1}{1 - i} = \frac{1 + i}{2}$. The integral is $2\pi i \times \frac{1 + i}{2} = \pi(i - 1) = (-1 + i)\pi$.