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25 Zeta and Related FunctionsRelated Functions

§25.12 Polylogarithms

Contents

§25.12(i) Dilogarithms

The notation Li2(z) was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):

25.12.1 Li2(z)=n=1znn2,
|z|1.
25.12.2 Li2(z)=-0zt-1ln(1-t)dt,
z\(1,).

Other notations and names for Li2(z) include S2(z) (Kölbig et al. (1970)), Spence function Sp(z) (’t Hooft and Veltman (1979)), and L2(z) (Maximon (2003)).

In the complex plane Li2(z) has a branch point at z=1. The principal branch has a cut along the interval [1,) and agrees with (25.12.1) when |z|1; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches.

25.12.3 Li2(z)+Li2(zz-1)=-12(ln(1-z))2,
z\[1,).
25.12.4 Li2(z)+Li2(1z)=-16π2-12(ln(-z))2,
z\[0,).
25.12.5 Li2(zm)=mk=0m-1Li2(ze2πik/m),
m=1,2,3,, |z|<1.
25.12.6 Li2(x)+Li2(1-x)=16π2-(lnx)ln(1-x),
0<x<1.

When z=eiθ, 0θ2π, (25.12.1) becomes

25.12.7 Li2(eiθ)=n=1cos(nθ)n2+in=1sin(nθ)n2.

The cosine series in (25.12.7) has the elementary sum

25.12.8 n=1cos(nθ)n2=π26-πθ2+θ24.

By (25.12.2)

25.12.9 n=1sin(nθ)n2=-0θln(2sin(12x))dx.

The right-hand side is called Clausen’s integral.

For graphics see Figures 25.12.1 and 25.12.2, and for further properties see Maximon (2003), Kirillov (1995), Lewin (1981), Nielsen (1909), and Zagier (1989).

See accompanying text
Figure 25.12.1: Dilogarithm function Li2(x), -20x<1. Magnify
Figure 25.12.2: Absolute value of the dilogarithm function |Li2(x+iy)|, -20x20, -20y20. Principal value. There is a cut along the real axis from 1 to . Magnify

§25.12(ii) Polylogarithms

For real or complex s and z the polylogarithm Lis(z) is defined by

25.12.10 Lis(z)=n=1znns.

For each fixed complex s the series defines an analytic function of z for |z|<1. The series also converges when |z|=1, provided that s>1. For other values of z, Lis(z) is defined by analytic continuation.

The notation ϕ(z,s) was used for Lis(z) in Truesdell (1945) for a series treated in Jonquière (1889), hence the alternative name Jonquière’s function. The special case z=1 is the Riemann zeta function: ζ(s)=Lis(1).

Integral Representation

25.12.11 Lis(z)=zΓ(s)0xs-1ex-zdx,

valid when s>0 and |ph(1-z)|<π, or s>1 and z=1. (In the latter case (25.12.11) becomes (25.5.1)).

Further properties include

25.12.12 Lis(z)=Γ(1-s)(ln1z)s-1+n=0ζ(s-n)(lnz)nn!,
s1,2,3,, |lnz|<2π,

and

25.12.13 Lis(e2πia)+eπisLis(e-2πia)=(2π)seπis/2Γ(s)ζ(1-s,a),

valid when s>0, a>0 or s>1, a=0. When s=2 and e2πia=z, (25.12.13) becomes (25.12.4).

See also Lewin (1981), Kölbig (1986), Maximon (2003), Prudnikov et al. (1990, §§1.2 and 2.5), Prudnikov et al. (1992a, §3.3), and Prudnikov et al. (1992b, §3.3).

§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

The Fermi–Dirac and Bose–Einstein integrals are defined by

25.12.14 Fs(x) =1Γ(s+1)0tset-x+1dt,
s>-1,
25.12.15 Gs(x) =1Γ(s+1)0tset-x-1dt,
s>-1, x<0; or s>0, x0,

respectively. Sometimes the factor 1/Γ(s+1) is omitted. See Cloutman (1989) and Gautschi (1993).

In terms of polylogarithms

25.12.16 Fs(x) =-Lis+1(-ex),
Gs(x) =Lis+1(ex).

For a uniform asymptotic approximation for Fs(x) see Temme and Olde Daalhuis (1990).