In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li_s(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral Li(z), which has the same notation without the subscript.

The polylogarithm function is defined by a power series in z generalizing the Mercator series, which is also a Dirichlet series in s: $${\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}=z+{z^{2} \over 2^{s}}+{z^{3} \over 3^{s}}+\cdots }$$

This definition is valid for arbitrary complex order s and for all complex arguments z with |z| < 1; it can be extended to |z| ≥ 1 by the process of analytic continuation. (Here the denominator k^s is understood as exp(s ln k)). The special case s = 1 involves the ordinary natural logarithm, Li₁(z) = −ln(1−z), while the special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral of itself: $${\displaystyle \operatorname {Li} _{s+1}(z)=\int _{0}^{z}{\frac {\operatorname {Li} _{s}(t)}{t}}dt}$$ thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function.

In the case where the order ${\displaystyle s}$ is an integer, it will be represented by ${\displaystyle s=n}$ (or ${\displaystyle s=-n}$ when negative). It is often convenient to define ${\displaystyle \mu =\ln(z)}$ where ${\displaystyle \ln(z)}$ is the principal branch of the complex logarithm ${\displaystyle \operatorname {Ln} (z)}$ so that ${\displaystyle -\pi <\operatorname {Im} (\mu )\leq \pi .}$ Also, all exponentiation will be assumed to be single-valued: ${\displaystyle z^{s}=\exp(s\ln(z)).}$

Depending on the order ${\displaystyle s}$, the polylogarithm may be multi-valued. The principal branch of ${\displaystyle \operatorname {Li} _{s}(z)}$ is taken to be given for ${\displaystyle |z|<1}$ by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from ${\displaystyle z=1}$ to ${\displaystyle \infty }$ such that the axis is placed on the lower half plane of ${\displaystyle z}$. In terms of ${\displaystyle \mu }$, this amounts to ${\displaystyle -\pi <\arg(-\mu )\leq \pi }$. The discontinuity of the polylogarithm in dependence on ${\displaystyle \mu }$ can sometimes be confusing.

For real argument ${\displaystyle z}$, the polylogarithm of real order ${\displaystyle s}$ is real if ${\displaystyle z<1}$, and its imaginary part for ${\displaystyle z\geq 1}$ is (Wood 1992, §3):

$${\displaystyle \operatorname {Im} \left(\operatorname {Li} _{s}(z)\right)=-{{\pi \mu ^{s-1}} \over {\Gamma (s)}}.}$$