GATE 2025 ME – Question 45
If $C$ is the unit circle in the complex plane with its center at the origin, then the value of $n$ in the equation given below is __________ (rounded off to 1 decimal place).
$$\oint_C \frac{z^3}{(z^2 + 4)(z^2 - 4)}\,dz = 2\pi\,i\,n$$
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Correct answer: -0.01 to 0.01
Explanation
The poles are at $z = \pm 2i$ and $z = \pm 2$, which all lie at distance 2 from the origin, outside the unit circle. The integrand has no singularity inside $C$, so by Cauchy's integral theorem the integral is 0, and $n = 0$.