GATE 2021 AE – Question 28
The counterclockwise contour integral $\oint_{|z|=1}z^3/(4z-i)\,dz$ is (round to three decimals)
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Correct answer: 0.015 to 0.035
Explanation
**Residue theorem.** For a counter-clockwise closed contour, $\oint f(z)\,dz=2\pi i\sum\text{Res}$ for the poles inside it.
**Pole.** $\dfrac{z^3}{4z-i}=\dfrac{z^3}{4\,(z-i/4)}$ has a simple pole at $z=i/4$, and $|i/4|=0.25<1$, so it lies inside the unit circle.
**Residue at $z=i/4$:**
$$\text{Res}=\frac{(i/4)^3}{4}=\frac{i^3/64}{4}=\frac{-i}{256}.$$
**Integral:**
$$\oint=2\pi i\cdot\frac{-i}{256}=\frac{2\pi}{256}=\frac{\pi}{128}=\mathbf{0.0245}\approx0.025.$$