GATE 2016 EC – Question 38
In the following integral, the contour $C$ encloses the points $2\pi j$ and $-2\pi j$.
$$-\frac{1}{2\pi} \oint_C \frac{\sin z}{(z - 2\pi j)^3} \, dz$$
The value of the integral is ________
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Correct answer: -136 to -132
Explanation
Only the pole at $z = 2\pi j$ matters. For a pole of order 3, the integral is $2\pi j \times \frac{1}{2!}\frac{d^2}{dz^2}\sin z \Big|_{z = 2\pi j} = 2\pi j \times \left(-\frac{\sin(2\pi j)}{2}\right)$. Since $\sin(2\pi j) = j\sinh(2\pi) = j(267.7)$, the integral is $2\pi j \times (-j \cdot 133.9) = 2\pi \times 133.9$. Multiplying by $-\frac{1}{2\pi}$ gives about $-133.9$.