Polylogarithm

For a complex order $s$ and $|z|<1$, the polylogarithm is the power seriesIt converges absolutely and locally uniformly in that disk. Differentiating term by term gives $z{\mathrm{Li}}_s^{\prime }(z)={\mathrm{Li}}_{s-1}(z)$. The apparent division by $z$ in the derivative formula has a removable value one at zero.

Two elementary cases are ${\mathrm{Li}}_0(z)=z/(1-z)$ and ${\mathrm{Li}}_1(z)=-\mathrm{Log}(1-z)$. The order-two case is the dilogarithm. At $z=1$, the defining sum is the Riemann zeta function when $\mathrm{Re}s>1$. Outside the disk, values are obtained by analytic continuation, with branch choices specified rather than inferred from a divergent series.^[ref]

The hierarchy is useful because differentiation lowers the order and integration raises it. Sums with powers of an integer denominator can therefore be related to logarithms, special values and Bose integrals.