Solution
= Solution
Factor the denominator as $z^2(1+z^2)$. Near zero,
$$
\frac1{z^2(1+z^2)}=z^{-2}-1+z^2-\cdots,
$$
so zero is a <double pole> with <residue> zero. At $z=\pm i$ the <poles> are simple and the denominator derivative is $2z+4z^3=-2z$. Therefore
$$
\boxed{\operatorname{Res}_{0}=0,\qquad\operatorname{Res}_{i}=\frac i2,\qquad\operatorname{Res}_{-i}=-\frac i2.}
$$
These are all the <poles>.