Dilogarithm Explained

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{Li} _{2(z)} = - \int ^{z{ ln ( 1- u)} _{0} \over u} du , z \in \Complex and its reflection.For, an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):\operatorname{Li} _{2(z)} = \sum ^{inf ty} _{k =1} {z ^{k} \over k ^{2}.}

Alternatively, the dilogarithm function is sometimes defined as

\int ^{v} _{1} (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 has hyperbolic volume

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

z = 1 , where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis ( 1 , inf ty) . 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) = (1)/(2)} \operatorname{Li} ^{2 ).} _{2 (z} ^[4] \operatorname{Li} _{2(1-z)+\operatorname{Li}} \right) =- _{2 \left( 1- (1)/(z)} (( ln z) ^{2})/(2) . \operatorname{Li} _{2(z)+\operatorname{Li}} ^{2 ( 1- z) = ({\pi)/(^{2}{6}-ln})} z ⋅ ln ( 1- z). ^[4] The reflection formula.\operatorname{Li} _{2(-z)-\operatorname{Li}} ^{2 ( 1- z) + (1)/(2)} \operatorname{Li} ^{2 ) =- ({\pi)/(^{2}{12}-ln})} _{2 ( 1- z} z ⋅ ln (z +1 ). \operatorname{Li} _{2(z)} + \operatorname{Li} _{2 \left( (1)/(z)} \right) = - (\pi ^{2})/(6) - (( ln ( - z)) ^{2})/(2) . ^[4] \operatorname{L}(z) + \operatorname{L}(y) = \operatorname{L}(xy) + \operatorname{L}( (x( 1- y))/() + \operatorname{L}( 1- xy) (y( 1- x))/(1- xy) ) .^{[5] [6]} Abel's functional equation or five-term relation where \operatorname{L}(x) = (\pi)/(6) [\operatorname{Li} _{2 (z) + (12ln (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( (1)/(3)} \right) - (1)/(6) \operatorname{Li} _{2 \left( (1)/(9)} \right) = ({\pi)/(^{2 }{ 18 } - (( ln3 ) ^{2})/(6)}) . \operatorname{Li} _{2 \left( - (1)/(3)} \right) - (1)/(3) \operatorname{Li} _{2 \left( (1)/(9)} \right) =- ({\pi)/(^{2 }{ 18 } + (( ln3 ) ^{2})/(6)}) . \operatorname{Li} _{2 \left( - (1)/(2)} \right) + (1)/(6) \operatorname{Li} _{2 \left( (1)/(9)} \right) =- ({\pi)/(ln3- ^{2}{18}+ln2 ⋅}) (( ln2 ) ^{2})/(- 2) (( ln3 ) ^{2})/(3) . \operatorname{Li} _{2 \left( (1)/(4)} \right) + (1)/(3) \operatorname{Li} _{2 \left( (1)/(9)} \right) = ({\pi)/(^{2}{18}+2ln2 ⋅ ln3-2(ln}) 2 ) ^{2- (2)/(3)} ( ln 3 ) ^{2.} \operatorname{Li} _{2 \left( - (1)/(8)} \right) + \operatorname{Li} _{2 \left( (1)/(9)} \right) =- (1)/(\left( ln { 2) (9)/(8) }\right)^2.36 \operatorname{Li} _{2 \left( (1)/(2)} \right) -36 \operatorname{Li} _{2 \left( (1)/(4)} \right) -12 \operatorname{Li} _{2 \left( (1)/(8)} \right) +6 \operatorname{Li} _{2 \left( (1)/(64)} \right) = {\pi} ^{2.}

Special values

\operatorname{Li} _{2 ( -1 ) =- ({\pi)/(^{2}{12}.})} \operatorname{Li} _{2(0)=0.} Its slope = 1.\operatorname{Li} _{2 \left( (1)/(2)} \right) = ({\pi)/(^{2 }{ 12 } - (( ln2 ) ^{2})/(2)}) . \operatorname{Li} _{2(1)} = \zeta( 2 ) = ({\pi)/(^{2}{6},}) where \zeta(s) is the Riemann zeta function.\operatorname{Li} _{2 ( 2 ) = ({\pi)/(^{2}{4}-i\piln2.})} \begin{align} \operatorname{Li} _{2 \left( - (\sqrt 5-1)/(2)} \right) & =- ({\pi)/(\left( ln ^{2 }{ 15 } + (1)/(2)}) (\sqrt 5+1)/(2) \right) ^{2} \\ & =- ({\pi)/(^{2 }{ 15 } + (1)/(2)}) \operatorname{arcsch} ^{2} 2 . \end{align} \begin{align} \operatorname{Li} _{2 \left( - (\sqrt 5+1)/(2)} \right) & =- ({\pi)/(^{2}{10}-ln}) ^{2} (\sqrt 5+1)/(\\ & =- 2) ({\pi)/(^{2}{10}-\operatorname{arcsch}}) ^{2} 2 . \end{align} \begin{align} \operatorname{Li} _{2 \left( (3- \sqrt 5)/(2)} \right) & = ({\pi)/(^{2}{15}-ln}) ^{2} (\sqrt 5+1)/(\\ & = 2) ({\pi)/(^{2}{15}-\operatorname{arcsch}}) ^{2} 2 . \end{align} \begin{align} \operatorname{Li} _{2 \left( (\sqrt 5-1)/(2)} \right) & = ({\pi)/(^{2}{10}-ln}) ^{2} (\sqrt 5+1)/(\\ & = 2) ({\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 ^{x} _{0} (ln | 1- u|)/(u) du = \begin{cases} \operatorname{Li} _{2(x),} & x \leq 1 ; \ (\pi ^{2})/(3) - (1)/(2) ( ln x) ^{2} - \operatorname{Li} _{2 ( (1)/(x)} ), & x > 1 . \end{cases}

See also

• Markstein number

References

• Book: Lewin . L. . Dilogarithms and associated functions . Macdonald . London . Foreword by J. C. P. Miller . 0105524 . 1958.
• Robert. Morris. Math. Comp.. 1979. The dilogarithm function of a real argument. 778–787. 33. 146. 10.1090/S0025-5718-1979-0521291-X . 521291. free.
• J. H.. Loxton. Special values of the dilogarithm. Acta Arith.. 1984. 18. 2. 155–166. 0736728. 10.4064/aa-43-2-155-166. free.
• Anatol N.. Kirillov. Dilogarithm identities. hep-th/9408113. 1995. 10.1143/PTPS.118.61. 118. Progress of Theoretical Physics Supplement. 61–142. 1995PThPS.118...61K. 119177149.
• Carlos. Osacar. Jesus. Palacian. Manuel. Palacios. Numerical evaluation of the dilogarithm of complex argument. 1995. 62. 1. 93–98. Celest. Mech. Dyn. Astron.. 10.1007/BF00692071. 1995CeMDA..62...93O. 121304484.
• Book: Zagier , Don . 2007 . The Dilogarithm Function . Frontiers in Number Theory, Physics, and Geometry II . Pierre Cartier . Pierre Moussa . Bernard Julia . Pierre Vanhove . 3–65 . 10.1007/978-3-540-30308-4_1 . 978-3-540-30308-4.

Further reading

• Book: Bloch, Spencer J. . Spencer Bloch

. Spencer Bloch . Higher regulators, algebraic K-theory, and zeta functions of elliptic curves . CRM Monograph Series . 11 . Providence, RI . . 2000 . 0-8218-2114-8 . 0958.19001 .

External links

• NIST Digital Library of Mathematical Functions: Dilogarithm

Notes and References

1. Zagier p. 10
2. Web site: William Spence - Biography.
3. Web site: Biography – GALT, JOHN – Volume VII (1836-1850) – Dictionary of Canadian Biography.
4. Zagier
5. Web site: Weisstein . Eric W. . Rogers L-Function . 2024-08-01 . mathworld.wolfram.com . en.
6. Rogers . L. J. . 1907 . On the Representation of Certain Asymptotic Series as Convergent Continued Fractions . Proceedings of the London Mathematical Society . en . s2-4 . 1 . 72–89 . 10.1112/plms/s2-4.1.72.