In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of 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 , which has the same notation without the subscript.
The polylogarithm function is defined by a power series in generalizing the Mercator series, which is also a Dirichlet series in :
This definition is valid for arbitrary complex order and for all complex arguments with ; it can be extended to by the process of analytic continuation. (Here the denominator is understood as ). The special case involves the ordinary natural logarithm, , while the special cases and 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:
thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders , the polylogarithm is a rational function.
Properties In the case where the order is an integer, it will be represented by (or when negative). It is often convenient to define where is the principal branch of the complex logarithm so that Also, all exponentiation will be assumed to be single-valued:
Depending on the order , the polylogarithm may be multi-valued. The principal branch of is taken to be given for by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from to such that the axis is placed on the lower half plane of In terms of this amounts to . The discontinuity of the polylogarithm in dependence on can sometimes be confusing.
For real argument , the polylogarithm of real order is real if and its imaginary part for is :
Going across the cut, if ε is an infinitesimally small positive real number, then:
Both can be concluded from the series expansion (see below) of about
The derivatives of the polylogarithm follow from the defining power series:
The square relationship is seen from the series definition, and is related to the duplication formula (see also , ):
Kummer's function obeys a very similar duplication formula. This is a special case of the multiplication formula, for any positive integer p:
which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. discrete Fourier transform).
Another important property, the inversion formula, involves the Hurwitz zeta function or the Bernoulli polynomials and is found under relationship to other functions below.
Particular values
For particular cases, the polylogarithm may be expressed in terms of other functions (see below). Particular values for the polylogarithm may thus also be found as particular values of these other functions.
For integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application of z·∂/∂z to Li1(z):
Accordingly the polylogarithm reduces to a ratio of polynomials in z, and is therefore a rational function of z, for all nonpositive integer orders. The general case may be expressed as a finite sum:
where S(n,k) are the Stirling numbers of the second kind. Equivalent formulae applicable to negative integer orders are :
and:
where are the Eulerian numbers. All roots of Li−n(z) are distinct and real; they include z = 0, while the remainder is negative and centered about z = −1 on a logarithmic scale. As n becomes large, the numerical evaluation of these rational expressions increasingly suffers from cancellation ; full accuracy can be obtained, however, by computing Li−n(z) via the general relation with the Hurwitz zeta function (see below). Some particular expressions for half-integer values of the argument z are:
where ζ is the Riemann zeta function. No formulae of this type are known for higher integer orders , but one has for instance :
which involves the alternating double sum
In general one has for integer orders n ≥ 2 :
where ζ(s1, …, sk) is the multiple zeta function; for example: As a straightforward consequence of the series definition, values of the polylogarithm at the pth complex roots of unity are given by the Fourier sum:
where ζ is the Hurwitz zeta function. For Re(s) > 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 , the polylogarithm reduces to the Riemann zeta function The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function: where is the Dirichlet eta function. For pure imaginary arguments, we have: where is the Dirichlet beta function. The polylogarithm is related to the complete Fermi–Dirac integral as: The polylogarithm is related to the complete Bose–Einstein 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 , and at 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 if , and for if . Equivalently, for all complex s and for complex , the inversion formula reads and for all complex s and for complex For , one has , and both expressions agree. These relations furnish the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series. (The corresponding equation of and is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously.) See the next item for a simplified formula when s is an integer. For positive integer polylogarithm orders s, the Hurwitz zeta function ζ(1−s, x) reduces to Bernoulli polynomials, , and Jonquière's inversion formula for n = 1, 2, 3, … becomes: where again 0 ≤ Re(x) < 1 if Im(x) ≥ 0, and 0 < Re(x) ≤ 1 if Im(x) < 0. Upon restriction of the polylogarithm argument to the unit circle, Im(x) = 0, the left hand side of this formula simplifies to 2 Re(Lin(e2πix)) if n is even, and to 2i Im(Lin(e2πix)) if n is odd. For negative integer orders, on the other hand, the divergence of Γ(s) implies for all z that : More generally, one has for : where both expressions agree for . (The corresponding equation of and is again not correct.) The polylogarithm with pure imaginary μ may be expressed in terms of the Clausen functions Cis(θ) and Sis(θ), and vice versa (; ): The inverse tangent integral can be expressed in terms of polylogarithms: The relation in particular implies: which explains the function name. The Legendre chi function χs(z) (; ) can be expressed as the odd part of polylogarithms: The polylogarithm of integer order can be expressed as a generalized hypergeometric function: In terms of the incomplete zeta functions or "Debye functions" : the polylogarithm Lin(z) for positive integer n may be expressed as the finite sum : A remarkably similar expression relates the "Debye functions" Zn(z) to the polylogarithm: Using Lambert series, if is Jordan's totient function, then
Integral representations Any of the following integral representations furnishes the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series. The polylogarithm can be expressed in terms of the integral of the Bose–Einstein 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 Bose integral but more commonly as a Bose–Einstein integral (, ).Bose integral is result of multiplication between Gamma function and Zeta function.
One can begin with equation for Bose integral, then use series equation.
Secondly, regroup expressions. Similarly, the polylogarithm can be expressed in terms of the integral of the Fermi–Dirac distribution:
This converges for and all 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. The papers of Dingle contain detailed investigations of both types of integrals.
The polylogarithm is also related to the integral of the Maxwell–Boltzmann distribution:
This also gives the asymptotic behavior of polylogarithm at the vicinity of origin. A complementary integral representation applies to Re(s) < 0 and to all z except to z real and ≥ 0:
This integral follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above) and a familiar integral representation of the latter. The polylogarithm may be quite generally represented by a Hankel contour integral , which extends the Bose–Einstein representation to negative orders s. As long as the t = μ pole of the integrand does not lie on the non-negative real axis, and s ≠ 1, 2, 3, …, we have:
where H represents the Hankel contour. The integrand has a cut along the real axis from zero to infinity, with the axis belonging to the lower half plane of t. The integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the poles t = μ + 2kπi, and terminates at +∞ on the lower half plane (Im(t) < 0). For the case where μ is real and non-negative, we can simply subtract the contribution of the enclosed t = μ pole:
where R is the residue of the pole: When the Abel–Plana formula is applied to the defining series of the polylogarithm, a Hermite-type integral representation results that is valid for all complex z and for all complex s:
where Γ is the upper incomplete gamma-function. All (but not part) of the ln(z) in this expression can be replaced by −ln(1⁄z). A related representation which also holds for all complex s,
avoids the use of the incomplete gamma function, but this integral fails for z on the positive real axis if Re(s) ≤ 0. This expression is found by writing 2s Lis(−z) / (−z) = Φ(z2, s, 1⁄2) − z Φ(z2, s, 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e2πt + 1) in place of 1 / (e2πt − 1) to the second Φ series. We can express an integral for the polylogarithm by integrating the ordinary geometric series termwise for as
Series representations