Spence's function

"Li2" redirects here. For the molecule with formula Li₂, see dilithium.

See also: polylogarithm#Dilogarithm

In mathematics, Spence's function, or dilogarithm, denoted as Li₂(z), is a particular case of the polylogarithm. Lobachevsky's function and Clausen's function are closely related functions. Two related special functions are referred to as Spence's function, the dilogarithm itself, and its reflection with the variable negated:

$\operatorname{Li}_2(\pm z) = -\int_0^z{\ln|1\mp\zeta| \over \zeta}\, \mathrm{d}\zeta = \sum_{k=1}^\infty {(\pm z)^k \over k^2};$

William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. He was at school with John Galt, who later wrote a biographical essay on Spence.

Identities

$\operatorname{Li}_2(-z)=-\operatorname{Li}_2\left(\frac{z}{1+z}\right)-\frac{\ln^2(1+z)}{2}$

$\operatorname{Li}_2({\rm{i}}z) =\frac{\operatorname{Li}_2(-z^2)}{4}+{\rm{i}} \operatorname{Li}_2(z)$

$\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2)$

$\operatorname{Li}_2(1-z)+\operatorname{Li}_2\left(1-\frac{1}{z}\right)=-\frac{\ln^2z}{2}$

$\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6}-\ln z \cdot\ln(1-z)$

$\operatorname{Li}_2(-z)-\operatorname{Li}_2(1-z)+\frac{1}{2}\operatorname{Li}_2(1-z^2)=-\frac {{\pi}^2}{12}-\ln z$

$\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18}-\ln^23$

$\operatorname{Li}_2\left(-\frac{1}{2}\right)+\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}-\ln2\cdot \ln3-\frac{\ln^22}{2}-\frac{\ln^23}{3}$

$\operatorname{Li}_2\left(\frac{1}{4}\right)+\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18}+2\ln2\ln3-2\ln^22-\frac{2}{3}\ln^23$

$\operatorname{Li}_2\left(-\frac{1}{3}\right)-\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\frac{1}{6}\ln^23$

$\operatorname{Li}_2\left(-\frac{1}{8}\right)+\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{1}{2}\ln^2{\frac{9}{8}}$

$36\operatorname{Li}_2\left(\frac{1}{2}\right)-36\operatorname{Li}_2\left(\frac{1}{4}\right)-12\operatorname{Li}_2\left(\frac{1}{8}\right)+6\operatorname{Li}_2\left(\frac{1}{64}\right)={\pi}^2$

Special values

$\operatorname{Li}_2(-1)=-\frac{{\pi}^2}{12}$  A072691

$\operatorname{Li}_2(0)=0$

$\operatorname{Li}_2\left(\frac{1}{2}\right)=\frac{{\pi}^2}{12}-\frac{\ln^2 2}{2}$  A076788

$\operatorname{Li}_2(1)=\frac{{\pi}^2}{6}$  A013661

$\operatorname{Li}_2(2)=\frac{{\pi}^2}{4}$  A091476

$\operatorname{Li}_2\left(-\frac{\sqrt5-1}{2}\right)=-\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5-1}{2}$

$=-\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2$

$\operatorname{Li}_2\left(-\frac{\sqrt5+1}{2}\right)=-\frac{{\pi}^2}{15}+\frac{1}{2}\ln^2 \frac{\sqrt5-1}{2}$  A152115

$=-\frac{{\pi}^2}{15}+\frac{1}{2}\operatorname{arcsch}^2 2$

$\operatorname{Li}_2\left(\frac{3+\sqrt5}{2}\right)=\frac{{\pi}^2}{15}-\frac{1}{2}\ln^2 \frac{\sqrt5-1}{2}$

$=\frac{{\pi}^2}{15}-\frac{1}{2}\operatorname{arcsch}^2 2$

$\operatorname{Li}_2\left(\frac{\sqrt5+1}{2}\right)=\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5-1}{2}$

$=\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2$

References

• Lewin, L. (1958), Dilogarithms and associated functions, Foreword by J. C. P. Miller, Macdonald, London, MR0105524 
• Morris, Robert (1979). "The dilogarithm function of a real argument". Math. Comp. 33 (146): 778–787. doi:10.1090/S0025-5718-1979-0521291-X. MR521291. 
• Kirillov, Anatol N. (1994). "Dilogarithm identities". arXiv:hep-th/9408113. 
• Osacar, Carlos; Palacian, Jesus; Palacios, Manuel (1995). "Numerical evaluation of the dilogarithm of complex argument". Celestial Mech. Dynam. Astron. 62 (1): 93–98. doi:10.1007/BF00692071. 
• Zagier, Don. "The Dilogarithm Function". http://maths.dur.ac.uk/~dma0hg/dilog.pdf. 
• Zagier, Don (2007). "The Dilogarithm Function". Front. Number Theory, Physics, Geom. II: 3–65. doi:10.1007/978-3-540-30308-4_1.