Dilogarithm

In mathematics, the dilogarithm (or Spence's function), 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:

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u},円du{\text{, }}z\in \mathbb {C} }${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u},円du{\text{, }}z\in \mathbb {C} }

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

${\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2}}.}${\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2}}.}

Alternatively, the dilogarithm function is sometimes defined as

${\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).}${\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).}

In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex whose vertices have cross ratio z has hyperbolic volume

${\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.}${\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.}

The function D(z) is sometimes called the Bloch-Wigner function.^[1] 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.^[2] He was at school with John Galt,^[3] who later wrote a biographical essay on Spence.

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

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2}}\operatorname {Li} _{2}(z^{2}).}${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2}}\operatorname {Li} _{2}(z^{2}).}^[4]

${\displaystyle \operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z}}\right)=-{\frac {(\ln z)^{2}}{2}}.}${\displaystyle \operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z}}\right)=-{\frac {(\ln z)^{2}}{2}}.}^[5]

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z).}${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z).}^[4] The reflection formula.

${\displaystyle \operatorname {Li} _{2}(-z)-\operatorname {Li} _{2}(1-z)+{\frac {1}{2}}\operatorname {Li} _{2}(1-z^{2})=-{\frac {{\pi }^{2}}{12}}-\ln z\cdot \ln(z+1).}${\displaystyle \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).}^[5]

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}\left({\frac {1}{z}}\right)=-{\frac {\pi ^{2}}{6}}-{\frac {(\ln(-z))^{2}}{2}}.}${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}\left({\frac {1}{z}}\right)=-{\frac {\pi ^{2}}{6}}-{\frac {(\ln(-z))^{2}}{2}}.}^[4]

${\displaystyle \operatorname {L} (x)+\operatorname {L} (y)=\operatorname {L} (xy)+\operatorname {L} ({\frac {x(1-y)}{1-xy}})+\operatorname {L} ({\frac {y(1-x)}{1-xy}})}${\displaystyle \operatorname {L} (x)+\operatorname {L} (y)=\operatorname {L} (xy)+\operatorname {L} ({\frac {x(1-y)}{1-xy}})+\operatorname {L} ({\frac {y(1-x)}{1-xy}})}.^{[6] [7]} Abel's functional equation or five-term relation where ${\displaystyle \operatorname {L} (z)={\frac {\pi }{6}}[\operatorname {Li} _{2}(z)+{\frac {1}{2}}\ln(z)\ln(1-z)]}${\displaystyle \operatorname {L} (z)={\frac {\pi }{6}}[\operatorname {Li} _{2}(z)+{\frac {1}{2}}\ln(z)\ln(1-z)]} is the Rogers L-function (an analogous relation is satisfied also by the quantum dilogarithm)

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{3}}\right)-{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}-{\frac {(\ln 3)^{2}}{6}}.}${\displaystyle \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}}.}^[5]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{3}}\right)-{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+{\frac {(\ln 3)^{2}}{6}}.}${\displaystyle \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}}.}^[5]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{2}}\right)+{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+\ln 2\cdot \ln 3-{\frac {(\ln 2)^{2}}{2}}-{\frac {(\ln 3)^{2}}{3}}.}${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{2}}\right)+{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+\ln 2\cdot \ln 3-{\frac {(\ln 2)^{2}}{2}}-{\frac {(\ln 3)^{2}}{3}}.}^[5]

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{4}}\right)+{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}+2\ln 2\cdot \ln 3-2(\ln 2)^{2}-{\frac {2}{3}}(\ln 3)^{2}.}${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{4}}\right)+{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}+2\ln 2\cdot \ln 3-2(\ln 2)^{2}-{\frac {2}{3}}(\ln 3)^{2}.} ^[5]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{8}}\right)+\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {1}{2}}\left(\ln {\frac {9}{8}}\right)^{2}.}${\displaystyle \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}.}^[5]

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

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

${\displaystyle \operatorname {Li} _{2}(0)=0.}${\displaystyle \operatorname {Li} _{2}(0)=0.} Its slope = 1.

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

${\displaystyle \operatorname {Li} _{2}(1)=\zeta (2)={\frac {{\pi }^{2}}{6}},}${\displaystyle \operatorname {Li} _{2}(1)=\zeta (2)={\frac {{\pi }^{2}}{6}},} where ${\displaystyle \zeta (s)}${\displaystyle \zeta (s)} is the Riemann zeta function.

${\displaystyle \operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}-i\pi \ln 2.}${\displaystyle \operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}-i\pi \ln 2.}

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}-1}{2}}\right)&=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\left(\ln {\frac {{\sqrt {5}}+1}{2}}\right)^{2}\\&=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\operatorname {arcsch} ^{2}2.\end{aligned}}}${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}-1}{2}}\right)&=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\left(\ln {\frac {{\sqrt {5}}+1}{2}}\right)^{2}\\&=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\operatorname {arcsch} ^{2}2.\end{aligned}}}

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}+1}{2}}\right)&=-{\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&=-{\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2.\end{aligned}}}${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}+1}{2}}\right)&=-{\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&=-{\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2.\end{aligned}}}

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac {3-{\sqrt {5}}}{2}}\right)&={\frac {{\pi }^{2}}{15}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&={\frac {{\pi }^{2}}{15}}-\operatorname {arcsch} ^{2}2.\end{aligned}}}${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac {3-{\sqrt {5}}}{2}}\right)&={\frac {{\pi }^{2}}{15}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&={\frac {{\pi }^{2}}{15}}-\operatorname {arcsch} ^{2}2.\end{aligned}}}

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac {{\sqrt {5}}-1}{2}}\right)&={\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&={\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2.\end{aligned}}}${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac {{\sqrt {5}}-1}{2}}\right)&={\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&={\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2.\end{aligned}}}

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:

${\displaystyle \operatorname {\Phi } (x)=-\int _{0}^{x}{\frac {\ln |1-u|}{u}},円du={\begin{cases}\operatorname {Li} _{2}(x),&x\leq 1;\\{\frac {\pi ^{2}}{3}}-{\frac {1}{2}}(\ln x)^{2}-\operatorname {Li} _{2}({\frac {1}{x}}),&x>1.\end{cases}}}${\displaystyle \operatorname {\Phi } (x)=-\int _{0}^{x}{\frac {\ln |1-u|}{u}},円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}}}

• Markstein number

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

• Special functions

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