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:

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

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

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

Alternatively, the dilogarithm function is sometimes defined as

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

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

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

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.

Analytic structure

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

Identities

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

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

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

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

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

Particular value identities

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

${\displaystyle {\mathrm{Li}}_2<!--\; \; -->\left(-<!--\; -\; -->\frac{1}{3}\right)-<!--\; -\; -->\frac{1}{3}{\mathrm{Li}}_2<!--\; \; -->\left(\frac{1}{9}\right)=-<!--\; -\; -->\frac{{\mathrm{\pi <!--\; \pi \; -->}}^2}{}}$Zdroj:https://en.wikipedia.org?pojem=Dilogarithm
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