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Discussion on question by onepound: How to compute the residue of $f(z)=\frac{n/z}{z^{n}-1}$ to be $-n$ at $z=0$?

Imported from a comment discussion on mathematica.stackexchange.com/questions/280534/…

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Discussion on question by onepound: How to compute the residue of $f(z)=\frac{n/z}{z^{n}-1}$ to be $-n$ at $z=0$?