Polylogarithm (${\mathrm{Li}}_s(z)$)

For $|z|<1$, the polylogarithm is ${\mathrm{Li}}_s(z)={\sum }_{n\ge 1}{z}^n/n^s$; contour formulas analytically continue it to a slit plane.

For $|z|<1$, $a\ge 0$ and integer $N\ge 0$, truncating the defining polylogarithm power series after $N$ terms has remainderThe inequality uses monotonicity of $k^a$ and then the geometric series. The bound applies to negative real $z$ as well as positive real $z$, and gives a rigorous absolute truncation tolerance independently of cancellation. It does not bound floating-point rounding error.

The dilogarithm is the order-two polylogarithm, defined for $|z|<1$ by ${\mathrm{Li}}_2(z)={\sum }_{n\ge 1}{z}^n/n^2$. Its derivative is $-\mathrm{Log}(1-z)/z$, with a removable value at zero; a chosen complex logarithm branch gives its analytic continuation.