Spence's function

The dilogarithm along the real axis

In mathematics, Spence's function, or dilogarithm, denoted as Li₂(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself:

$\operatorname{Li}_2(z) = -\int_0^z{\ln(1-u) \over u}\, \mathrm{d}u \text{, }z \in\mathbb{C} \setminus [1,\infty)$

and its reflection. For $|z|<1$ an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):

$\operatorname{Li}_2(z) = \sum_{k=1}^\infty {z^k \over k^2}.$

Alternatively, the dilogarithm function is sometimes defined as

$\int_{1}^{v} \frac{ \ln t }{ 1 -t } \mathrm{d}t = \operatorname{Li}_2(1-v).$

In hyperbolic geometry the dilogarithm $\operatorname{Li}_2(z)$ occurs as the hyperbolic volume of an ideal simplex whose ideal vertices have cross ratio $z$. Lobachevsky's function and Clausen's function are closely related functions.

William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century.^[1] He was at school with John Galt,^[2] who later wrote a biographical essay on Spence.

Identities

$\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2)$^[3]

$\operatorname{Li}_2(1-z)+\operatorname{Li}_2\left(1-\frac{1}{z}\right)=-\frac{\ln^2z}{2}$^[4]

$\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6}-\ln z \cdot\ln(1-z)$^[3]

$\operatorname{Li}_2(-z)-\operatorname{Li}_2(1-z)+\frac{1}{2}\operatorname{Li}_2(1-z^2)=-\frac {{\pi}^2}{12}-\ln z \cdot \ln(z+1)$^[4]

$\operatorname{Li}_2(z) +\operatorname{Li}_2(\frac{1}{z}) = - \frac{\pi^2}{6} - \frac{1}{2}\ln^2(-z)$^[3]

Particular value identities

$\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18}-\frac{\ln^23}{6}$^[4]

$\operatorname{Li}_2\left(-\frac{1}{2}\right)+\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\ln2\cdot \ln3-\frac{\ln^22}{2}-\frac{\ln^23}{3}$^[4]

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

$\operatorname{Li}_2\left(-\frac{1}{3}\right)-\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\frac{1}{6}\ln^23$ ^[4]

$\operatorname{Li}_2\left(-\frac{1}{8}\right)+\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{1}{2}\ln^2{\frac{9}{8}}$^[4]

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

Special values

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

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

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

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

$\operatorname{Li}_2(2)=\frac{{\pi}^2}{4}-i\pi\ln2$

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

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

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

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

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

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

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

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

In Particle Physics

Spence's Function is commonly encountered in particle physics while calculating radiative corrections. In this context, the function is often defined with an absolute value inside the logarithm:

$\operatorname{\Phi}(x) = -\int_0^x \frac{\ln|1-u|}{u} \, \mathrm{d}u = \begin{cases} \operatorname{Li}_2(x), & x \leq 1; \\ \frac{\pi^2}{3} - \frac{1}{2} \ln^2(x) - \operatorname{Li}_2(\frac{1}{x}), & x > 1. \end{cases}$

Notes

1. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Spence.html
2. http://www.biographi.ca/009004-119.01-e.php?BioId=37522
3. Zagier
4. Weisstein, Eric W., "Dilogarithm", MathWorld.

References

• Lewin, L. (1958). Dilogarithms and associated functions. Foreword by J. C. P. Miller. London: Macdonald. MR 0105524.
• Morris, Robert (1979). "The dilogarithm function of a real argument". Math. Comp. 33 (146): 778–787. doi:10.1090/S0025-5718-1979-0521291-X. MR 521291.
• Loxton, J. H. (1984). "Special values of the dilogarithm". Acta Arith. 18 (2): 155–166. MR 0736728.
• 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 (2007). "The Dilogarithm Function" (PDF). Front. Number Theory, Physics, Geom. II: 3–65. doi:10.1007/978-3-540-30308-4_1.

Further reading

• Bloch, Spencer J. (2000). Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.

External links

• NIST Digital Library of Mathematical Functions: Dilogarithm
• Weisstein, Eric W., "Dilogarithm", MathWorld.