Solution
= Solution
When $s$ is a nonpositive integer, $t^{s-1}$ is meromorphic and the Hankel contour may be shrunk to a small circle about zero. The integral is then computed by the residue at zero. This gives
$$
\boxed{\operatorname{Li}_0(z)=\frac z{1-z}},
$$
$$
\boxed{\operatorname{Li}_{-1}(z)=\frac z{(1-z)^2}},
$$
$$
\boxed{\operatorname{Li}_{-2}(z)=\frac{z(1+z)}{(1-z)^3}}.
$$
These also follow by repeatedly applying $z\,d/dz$ to the geometric series.