GATE 2017 EC (EC2) – Question 14
The residues of the function $f(z)=1/[(z-4)(z+1)^3]$ are
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Correct answer: (B) $1/125$ and $-1/125$
Explanation
There are poles at $z=4$ and $z=-1$. At the simple pole, the residue is $\lim_{z\to4}(z-4)f(z)=1/5^3=1/125$.
At the third-order pole, differentiate the analytic factor twice and divide by $2!$:
$$\operatorname{Res}_{-1}f=\frac1{2!}\left[\frac{d^2}{dz^2}\frac1{z-4}\right]_{z=-1}.$$
Since the second derivative is $2/(z-4)^3$, the residue is $-1/125$. The pair is **B**.