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Dilogarithm

Special function

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:

Li 2 ⁡ ( z ) = − ∫ 0 z ln ⁡ ( 1 − u ) u d u , z ∈ C {\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):

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

Alternatively, the dilogarithm function is sometimes defined as

∫ 1 v ln ⁡ t 1 − t d t = 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

D ( z ) = Im ⁡ 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. 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.

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Analytic structure

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

Identities

Li 2 ⁡ ( z ) + Li 2 ⁡ ( − z ) = 1 2 Li 2 ⁡ ( z 2 ) . {\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2}}\operatorname {Li} _{2}(z^{2}).} ⁴ Li 2 ⁡ ( 1 − z ) + Li 2 ⁡ ( 1 − 1 z ) = − ( 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 Li 2 ⁡ ( z ) + Li 2 ⁡ ( 1 − z ) = π 2 6 − ln ⁡ z ⋅ ln ⁡ ( 1 − z ) . {\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z).} ⁶ The reflection formula. Li 2 ⁡ ( − z ) − Li 2 ⁡ ( 1 − z ) + 1 2 Li 2 ⁡ ( 1 − z 2 ) = − π 2 12 − ln ⁡ z ⋅ 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).} ⁷ Li 2 ⁡ ( z ) + Li 2 ⁡ ( 1 z ) = − π 2 6 − ( 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}}.} ⁸ L ⁡ ( x ) + L ⁡ ( y ) = L ⁡ ( x y ) + L ⁡ ( x ( 1 − y ) 1 − x y ) + L ⁡ ( y ( 1 − x ) 1 − x y ) {\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}})} .⁹ ¹⁰ Abel's functional equation or five-term relation where L ⁡ ( z ) = π 6 [ Li 2 ⁡ ( z ) + 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)

Particular value identities

Li 2 ⁡ ( 1 3 ) − 1 6 Li 2 ⁡ ( 1 9 ) = π 2 18 − ( 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}}.} ¹¹ Li 2 ⁡ ( − 1 3 ) − 1 3 Li 2 ⁡ ( 1 9 ) = − π 2 18 + ( 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}}.} ¹² Li 2 ⁡ ( − 1 2 ) + 1 6 Li 2 ⁡ ( 1 9 ) = − π 2 18 + ln ⁡ 2 ⋅ ln ⁡ 3 − ( ln ⁡ 2 ) 2 2 − ( 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}}.} ¹³ Li 2 ⁡ ( 1 4 ) + 1 3 Li 2 ⁡ ( 1 9 ) = π 2 18 + 2 ln ⁡ 2 ⋅ ln ⁡ 3 − 2 ( ln ⁡ 2 ) 2 − 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}.} ¹⁴ Li 2 ⁡ ( − 1 8 ) + Li 2 ⁡ ( 1 9 ) = − 1 2 ( ln ⁡ 9 8 ) 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}.} ¹⁵ 36 Li 2 ⁡ ( 1 2 ) − 36 Li 2 ⁡ ( 1 4 ) − 12 Li 2 ⁡ ( 1 8 ) + 6 Li 2 ⁡ ( 1 64 ) = π 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}.}

Special values

Li 2 ⁡ ( − 1 ) = − π 2 12 . {\displaystyle \operatorname {Li} _{2}(-1)=-{\frac {{\pi }^{2}}{12}}.} Li 2 ⁡ ( 0 ) = 0. {\displaystyle \operatorname {Li} _{2}(0)=0.} Its slope = 1. Li 2 ⁡ ( 1 2 ) = π 2 12 − ( ln ⁡ 2 ) 2 2 . {\displaystyle \operatorname {Li} _{2}\left({\frac {1}{2}}\right)={\frac {{\pi }^{2}}{12}}-{\frac {(\ln 2)^{2}}{2}}.} Li 2 ⁡ ( 1 ) = ζ ( 2 ) = π 2 6 , {\displaystyle \operatorname {Li} _{2}(1)=\zeta (2)={\frac {{\pi }^{2}}{6}},} where ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. Li 2 ⁡ ( 2 ) = π 2 4 − i π ln ⁡ 2. {\displaystyle \operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}-i\pi \ln 2.} Li 2 ⁡ ( − 5 − 1 2 ) = − π 2 15 + 1 2 ( ln ⁡ 5 + 1 2 ) 2 = − π 2 15 + 1 2 arcsch 2 ⁡ 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}}} Li 2 ⁡ ( − 5 + 1 2 ) = − π 2 10 − ln 2 ⁡ 5 + 1 2 = − π 2 10 − arcsch 2 ⁡ 2. {\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}}} Li 2 ⁡ ( 3 − 5 2 ) = π 2 15 − ln 2 ⁡ 5 + 1 2 = π 2 15 − arcsch 2 ⁡ 2. {\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}}} Li 2 ⁡ ( 5 − 1 2 ) = π 2 10 − ln 2 ⁡ 5 + 1 2 = π 2 10 − arcsch 2 ⁡ 2. {\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}}}

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:

Φ ⁡ ( x ) = − ∫ 0 x ln ⁡ | 1 − u | u d u = { Li 2 ⁡ ( x ) , x ≤ 1 ; π 2 3 − 1 2 ( ln ⁡ x ) 2 − Li 2 ⁡ ( 1 x ) , x > 1. {\displaystyle \operatorname {\Phi } (x)=-\int _{0}^{x}{\frac {\ln |1-u|}{u}}\,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}}}

Notes

Lewin, L. (1958). Dilogarithms and associated functions. Foreword by J. C. P. Miller. London: Macdonald. MR 0105524.
Morris, Robert (1979). "The dilogarithm function of a real argument". Math. Comp. 33 (146): 778–787. doi:10.1090/S0025-5718-1979-0521291-X. MR 0521291.
Loxton, J. H. (1984). "Special values of the dilogarithm". Acta Arith. 18 (2): 155–166. doi:10.4064/aa-43-2-155-166. MR 0736728.
Kirillov, Anatol N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement. 118: 61–142. arXiv:hep-th/9408113. Bibcode:1995PThPS.118...61K. doi:10.1143/PTPS.118.61. S2CID 119177149.
Osacar, Carlos; Palacian, Jesus; Palacios, Manuel (1995). "Numerical evaluation of the dilogarithm of complex argument". Celest. Mech. Dyn. Astron. 62 (1): 93–98. Bibcode:1995CeMDA..62...93O. doi:10.1007/BF00692071. S2CID 121304484.
Zagier, Don (2007). "The Dilogarithm Function". In Pierre Cartier; Pierre Moussa; Bernard Julia; Pierre Vanhove (eds.). Frontiers in Number Theory, Physics, and Geometry II (PDF). pp. 3–65. doi:10.1007/978-3-540-30308-4_1. ISBN 978-3-540-30308-4.

External links

References

Zagier p. 10 ↩
"William Spence - Biography". https://mathshistory.st-andrews.ac.uk/Biographies/Spence/ ↩
"Biography – GALT, JOHN – Volume VII (1836-1850) – Dictionary of Canadian Biography". http://www.biographi.ca/009004-119.01-e.php?BioId=37522 ↩
Zagier ↩
Weisstein, Eric W. "Dilogarithm". MathWorld. /wiki/Eric_W._Weisstein ↩
Zagier ↩
Weisstein, Eric W. "Dilogarithm". MathWorld. /wiki/Eric_W._Weisstein ↩
Zagier ↩
Weisstein, Eric W. "Rogers L-Function". mathworld.wolfram.com. Retrieved 2024-08-01. https://mathworld.wolfram.com/ ↩
Rogers, L. J. (1907). "On the Representation of Certain Asymptotic Series as Convergent Continued Fractions". Proceedings of the London Mathematical Society. s2-4 (1): 72–89. doi:10.1112/plms/s2-4.1.72. http://doi.wiley.com/10.1112/plms/s2-4.1.72 ↩
Weisstein, Eric W. "Dilogarithm". MathWorld. /wiki/Eric_W._Weisstein ↩
Weisstein, Eric W. "Dilogarithm". MathWorld. /wiki/Eric_W._Weisstein ↩
Weisstein, Eric W. "Dilogarithm". MathWorld. /wiki/Eric_W._Weisstein ↩
Weisstein, Eric W. "Dilogarithm". MathWorld. /wiki/Eric_W._Weisstein ↩
Weisstein, Eric W. "Dilogarithm". MathWorld. /wiki/Eric_W._Weisstein ↩

Analytic structure

Identities

Particular value identities

Special values

In particle physics

See also

Notes

Further reading

External links

References