5.79. Determine the Laurent series expansion of the function $f$ in the
specified ring $D$: (i) $f(z) = (z^2 + 1)^{-1}$, $D = \{z: 1 < |z - 2i| < 3\}$;
(ii) $f(z) = z^{-4}(e^{z^2} - 1)$, $D = \Delta^*(0, \infty)$; (iii) $f(z) = (z - 1)^{-2} \log z$,
$D = \Delta^*(1, 1)$; (iv) $f(z) = (z^2 - 4z)^{-1}$, $D = \{z: 3 < |z - 3i| < 5\}$;
(v) $f(z) = z(z - 2)^{-4} \cos \pi z$, $D = \Delta^*(2, \infty)$; (vi) $f(z) = (z - 1)\sin(z^{-1})$,
$D = \Delta^*(0, \infty)$; (vii) $f(z) = z^6 \cos^2(z^{-2})$, $D = \Delta^*(0, \infty)$.
