Discussion
True or false: the residue at a pole of order two can be read off as the limit of the function times the vanishing factor.
Answer: C
For a pole of order two the recipe is
\[ \mathrm{Res}_{z=a} f \;=\; \lim_{z \to a} \frac{d}{dz}\left[(z-a)^{2}f(z)\right], \]
so the squared factor removes the pole and the differentiation extracts the coefficient \( a_{-1} \) rather than \( a_{-2} \). A double pole does have a residue, and it may well be complex
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