Dilogarithm

Analytic structure

Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at z = 1, where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis (1, \infty). However, the function is continuous at the branch point and takes on the value \operatorname{Li}_2(1) = \pi^2/6.

Identities

:\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 z)^2}{2}. :\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6}-\ln z \cdot\ln(1-z). The reflection formula. :\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). :\operatorname{Li}_2(z) +\operatorname{Li}_2\left(\frac{1}{z}\right) = - \frac{\pi^2}{6} - \frac{(\ln(-z))^2}{2}. :\operatorname{L}(x)+\operatorname{L}(y)=\operatorname{L}(xy)+\operatorname{L}\left(\frac{x(1-y)}{1-xy}\right)+\operatorname{L}\left(\frac{y(1-x)}{1-xy}\right). Abel's functional equation or five-term relation where \operatorname{L}(z)=\frac{\pi}{6}[\operatorname{Li}_2(z)+\frac12\ln(z)\ln(1-z)] is the Rogers L-function (an analogous relation is satisfied also by the quantum dilogarithm)

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 3)^2}{6}. :\operatorname{Li}_2\left(-\frac{1}{3}\right)-\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\frac{(\ln 3)^2}{6}. :\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 2)^2}{2}-\frac{(\ln 3)^2}{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\cdot\ln3-2(\ln 2)^2-\frac{2}{3}(\ln 3)^2. :\operatorname{Li}_2\left(-\frac{1}{8}\right)+\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{1}{2}\left(\ln{\frac{9}{8}}\right)^2. :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. Its slope = 1. :\operatorname{Li}_2\left(\frac{1}{2}\right)=\frac{{\pi}^2}{12}-\frac{(\ln 2)^2}{2}. :\operatorname{Li}_2(1) = \zeta(2) = \frac{{\pi}^2}{6}, where \zeta(s) is the Riemann zeta function. :\operatorname{Li}_2(2)=\frac{{\pi}^2}{4}-i\pi\ln2. :\begin{align} \operatorname{Li}_2\left(-\frac{\sqrt5-1}{2}\right) &=-\frac{{\pi}^2}{15}+\frac{1}{2}\left(\ln\frac{\sqrt5+1}{2}\right)^2 \ &=-\frac{{\pi}^2}{15}+\frac{1}{2}\operatorname{arcsch}^2 2. \end{align} :\begin{align} \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. \end{align} :\begin{align} \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. \end{align} :\begin{align} \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. \end{align}

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} , du = \begin{cases} \operatorname{Li}_2(x), & x \leq 1; \ \frac{\pi^2}{3} - \frac{1}{2}(\ln x)^2 - \operatorname{Li}_2(\frac{1}{x}), & x 1. \end{cases}

Notes

References

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References

1. Zagier p. 10
2. "William Spence - Biography".
3. "Biography – GALT, JOHN – Volume VII (1836-1850) – Dictionary of Canadian Biography".
4. Zagier
5. "Dilogarithm".
6. Weisstein, Eric W.. "Rogers L-Function".
7. Rogers, L. J.. (1907). "On the Representation of Certain Asymptotic Series as Convergent Continued Fractions". Proceedings of the London Mathematical Society.