# dilogarithms

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## 1—10 of 13 matching pages

##### 1: 25.19 Tables
• Abramowitz and Stegun (1964) tabulates: $\zeta\left(n\right)$, $n=2,3,4,\dots$, 20D (p. 811); $\operatorname{Li}_{2}\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta)$, $\theta=15^{\circ}(1^{\circ})30^{\circ}(2^{\circ})90^{\circ}(5^{\circ})180^{\circ}$, $f(\theta)+\theta\ln\theta$, $\theta=0(1^{\circ})15^{\circ}$, 6D (p. 1006). Here $f(\theta)$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

• Morris (1979) tabulates $\operatorname{Li}_{2}\left(x\right)$25.12(i)) for $\pm x=0.02(.02)1(.1)6$ to 30D.

• Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

• Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

• ##### 2: 25.12 Polylogarithms
###### §25.12(i) Dilogarithms
The notation $\operatorname{Li}_{2}\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … Other notations and names for $\operatorname{Li}_{2}\left(z\right)$ include $S_{2}(z)$ (Kölbig et al. (1970)), Spence function $\mathrm{Sp}(z)$ (’t Hooft and Veltman (1979)), and $\mathrm{L}_{2}(z)$ (Maximon (2003)). In the complex plane $\operatorname{Li}_{2}\left(z\right)$ has a branch point at $z=1$. … 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). …
##### 3: 25.18 Methods of Computation
For the Hurwitz zeta function $\zeta\left(s,a\right)$ see Spanier and Oldham (1987, p. 653) and Coffey (2009). For dilogarithms and polylogarithms see Jacobs and Lambert (1972), Osácar et al. (1995), Spanier and Oldham (1987, pp. 231–232), and Zudilin (2007). …
##### 4: 25.20 Approximations
• Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta\left(s\right)$ for $0\leq s\leq 1$ (15D), $\zeta\left(s+1\right)$ for $0\leq s\leq 1$ (20D), and $\ln\xi\left(\tfrac{1}{2}+ix\right)$25.4) for $-1\leq x\leq 1$ (20D). For errata see Piessens and Branders (1972).

• Morris (1979) gives rational approximations for $\operatorname{Li}_{2}\left(x\right)$25.12(i)) for $0.5\leq x\leq 1$. Precision is varied with a maximum of 24S.

• ##### 5: 25.1 Special Notation
The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\operatorname{Li}_{2}\left(z\right)$, the polylogarithm $\operatorname{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 7: Bibliography Z
• D. Zagier (1989) The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.
##### 9: Bibliography K
• A. N. Kirillov (1995) Dilogarithm identities. Progr. Theoret. Phys. Suppl. (118), pp. 61–142.
• K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.
• ##### 10: Bibliography O
• C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.