In mathematics, the polylogarithm is a special function Lis(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 rational functions. 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 which has a similar notation.
The polylogarithm function is defined by the infinite sum, or power series:
This definition is valid for arbitrary complex order s and for all complex
arguments z with |z| 1, where Lis(1) is finite, the relation also holds with m =
0 or m = p. While this formula is not as simple as that implied by the more
general relation with the Hurwitz zeta function listed under relationship to
other functions below, it has the advantage of applying to non-negative
integer values of s as well. As usual, the relation may be inverted to express
ζ(s, m⁄p) for any m = 1, ..., p as a Fourier sum of Lis(exp(2πi k⁄p)) over k
= 1, ..., p. Relationship to other functions
For z = 1 the polylogarithm reduces to the Riemann zeta function
The polylogarithm is related to Dirichlet eta function and the Dirichlet
beta function: where η(s) is the Dirichlet eta
function. For pure imaginary arguments, we have:
where β(s) is the Dirichlet beta function.
The polylogarithm is related to the complete Fermi–Dirac integral as:
The polylogarithm is a special case of the incomplete polylogarithm function
The polylogarithm is a special case of the Lerch transcendent
The polylogarithm is related to the Hurwitz zeta function by:
which relation, however, is invalidated at positive integer s by poles of the
gamma function Γ(1−s), and at s = 0 by a pole of both zeta functions; a
derivation of this formula is given under series representations below. With
a little help from a functional equation for the Hurwitz zeta function, the
polylogarithm is consequently also related to that function via:
which relation holds for 0 ≤ Re(x) 0 and all z except for z real and ≥ 1. The
polylogarithm in this context is sometimes referred to as a Bose integral
or a Bose–Einstein integral. Similarly, the polylogarithm can be expressed in
terms of the integral of the Fermi–Dirac distribution:
This converges for Re(s) > 0 and all z except for z real and ≤ −1. The
polylogarithm in this context is sometimes referred to as a Fermi
integral or a Fermi–Dirac integral. These representations are readily
verified by Taylor expansion of the integrand with respect to z and termwise
integration. 2. A complementary integral
representation applies to Re(s) 0), circles the origin without enclosing any
of the poles t = µ + 2kπi, and terminates at +∞ on the lower half plane