Spence's function

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"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[edit]

\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[edit]

\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[edit]

\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}-\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

In Nuclear Physics[edit]

Spence's Function is commonly encountered in nuclear 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{\ln|1-u| \over u}\mathrm{d}u

Notes[edit]

References[edit]

Further reading[edit]

External links[edit]