Spence's function

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

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}\left({\frac {1}{z}}\right)=-{\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

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". Celest. Mech. Dyn. 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.

External links

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Spence's function

Identities

Particular value identities

Special values

In Particle Physics

Notes

References

Further reading

External links