Questions tagged [polylogarithm]

The polylogarithm function $\operatorname{Li}_s(z)$ is defined by the infinite sum

$$\operatorname{Li}_s(z)=\sum_{k = 1}^{\infty} \frac{z^k}{k^s}$$

for all $|z| < 1$ and complex order $s$, and obtained by analytic continuation of the sum. Depending on the order $s$, a branch cut must be taken for the logarithm.

In particular cases, the polylogarithm may have simpler representations; for example,

\begin{align*} \operatorname{Li}_1(z) &= -\ln{(1 - z)} \\ \operatorname{Li}_0(z) &= \frac{z}{1 - z} \\ ...\ &=\ ...\\ \operatorname{Li}_{n-1}(z)&=z\frac{\mathrm{d}}{\mathrm{d}z}\text{Li}_n(z) \end{align*}

In the cases $s = 2$ and $s = 3$, the function is called the dilogarithm and trilogarithm, respectively.

The polylogarithm functions arise in quantum statistics and electrodynamics, and are related to the Fermi-Dirac integral.

Is directly related to the riemann-zeta function: $$\text{Li}_s(1)=\zeta(s)\qquad \text{for }s>1$$

Furthermore, its derivatives with respect to the parameter $s$ are also defined:

$$\text{Li}_s^{(n,0)}(z):=\frac{\partial^n}{\partial s^{s}}\text{Li}_s(z)=\sum_{k=1}^{\infty}(-1)^n\sum_{k=1}^{\infty}\frac{\ln(k)^n}{k^s}z^k\qquad\text{for }|z|<1$$