Discussion
Find the residue of \( \dfrac{1}{z^{2} - 1} \) at \( z = 1 \).
Answer: C
Factorise and take the limit:
\[ \mathrm{Res}_{z=1} \;=\; \lim_{z \to 1} (z-1)\cdot\frac{1}{(z-1)(z+1)} \;=\; \frac{1}{1 + 1} \;=\; \frac{1}{2}. \]
The residue at the other pole \( z = -1 \) works out at \( -1/2 \), and the two sum to zero, which is why a contour enclosing both gives a vanishing integral
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