§25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions (original) (raw)

Contents

1. §25.12(i) Dilogarithms
2. §25.12(ii) Polylogarithms
3. §25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

§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):

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.

When z=ei⁢θ, 0≤θ≤2⁢π, (25.12.1) becomes

The cosine series in (25.12.7) has the elementary sum

By (25.12.2)

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).

Figure 25.12.1: Dilogarithm function Li2⁡(x),−20≤x<1. Magnify 3D Help

Figure 25.12.2: Absolute value of the dilogarithm function |Li2⁡(x+i⁢y)|,−20≤x≤20,−20≤y≤20. Principal value. There is a cut along the real axis from 1 to ∞.Magnify 3D Help

§25.12(ii) Polylogarithms

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

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

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

Integral Representation

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

Further properties include

and

When s=2 (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

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

In terms of polylogarithms

For a uniform asymptotic approximation for Fs⁡(x) seeTemme and Olde Daalhuis (1990).