Dilogarithm

In mathematics, the dilogarithm (or Spence's function), denoted as , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself:
:$\operatorname_2(z) = -\int_0^z\, du \textz \in \Complex$
and its reflection.
For , an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):
:$\operatorname_2(z) = \sum_^\infty .$

Alternatively, the dilogarithm function is sometimes defined as
:$\int_^ \frac dt = \operatorname_2(1-v).$

In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex whose vertices have cross ratio has hyperbolic volume
:$D(z) = \operatorname \operatorname_2(z) + \arg(1-z) \log, z, .$
The function is sometimes called the Bloch-Wigner function. 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. He was at school with John Galt, 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 $z = 1$, 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 $\operatorname_2(1) = \pi^2/6$.

Identities

:$\operatorname_2(z)+\operatorname_2(-z)=\frac\operatorname_2(z^2).$Zagier
:$\operatorname_2(1-z)+\operatorname_2\left(1-\frac\right)=-\frac.$
:$\operatorname_2(z)+\operatorname_2(1-z)=\frac-\ln z \cdot\ln(1-z).$ The reflection formula.
:$\operatorname_2(-z)-\operatorname_2(1-z)+\frac\operatorname_2(1-z^2)=-\frac-\ln z \cdot \ln(z+1).$
:$\operatorname_2(z) +\operatorname_2\left(\frac\right) = - \frac - \frac.$
:$\operatorname(x)+\operatorname(y)=\operatorname(xy)+\operatorname(\frac)+\operatorname(\frac)$. Abel's functional equation or five-term relation where \operatorname(z)=\frac operatorname_2(z)+\frac12\ln(z)\ln(1-z)/math> is the Rogers L-function (an analogous relation is satisfied also by the quantum dilogarithm)

Particular value identities

:$\operatorname_2\left(\frac\right)-\frac\operatorname_2\left(\frac\right)=\frac-\frac.$
:$\operatorname_2\left(-\frac\right)-\frac\operatorname_2\left(\frac\right)=-\frac+\frac.$
:$\operatorname_2\left(-\frac\right)+\frac\operatorname_2\left(\frac\right)=-\frac+\ln2\cdot \ln3-\frac-\frac.$
:$\operatorname_2\left(\frac\right)+\frac\operatorname_2\left(\frac\right)=\frac+2\ln2\cdot\ln3-2(\ln 2)^2-\frac(\ln 3)^2.$
:$\operatorname_2\left(-\frac\right)+\operatorname_2\left(\frac\right)=-\frac\left(\ln\right)^2.$
:$36\operatorname_2\left(\frac\right)-36\operatorname_2\left(\frac\right)-12\operatorname_2\left(\frac\right)+6\operatorname_2\left(\frac\right)=^2.$

Special values

:$\operatorname_2(-1)=-\frac.$
:$\operatorname_2(0)=0.$ Its slope = 1.
:$\operatorname_2\left(\frac\right)=\frac-\frac.$
:$\operatorname_2(1) = \zeta(2) = \frac,$ where $\zeta(s)$ is the Riemann zeta function.
:$\operatorname_2(2)=\frac-i\pi\ln2.$
:$\begin \operatorname_2\left(-\frac\right) &=-\frac+\frac\left(\ln\frac\right)^2 \\ &=-\frac+\frac\operatorname^2 2. \end$
:$\begin \operatorname_2\left(-\frac\right) &=-\frac-\ln^2 \frac \\ &=-\frac-\operatorname^2 2. \end$
:$\begin \operatorname_2\left(\frac\right) &=\frac-\ln^2 \frac \\ &=\frac-\operatorname^2 2. \end$
:$\begin \operatorname_2\left(\frac\right) &=\frac-\ln^2 \frac \\ &=\frac-\operatorname^2 2. \end$

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:

:$\operatorname(x) = -\int_0^x \frac \, du = \begin \operatorname_2(x), & x \leq 1; \\ \frac - \frac(\ln x)^2 - \operatorname_2(\frac), & x > 1. \end$

See also

* Markstein number

Notes

References

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Further reading

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External links

NIST Digital Library of Mathematical Functions: Dilogarithm
* {{MathWorld, title=Dilogarithm, urlname=Dilogarithm

Special functions