GeneralizedPolylog - Maple Help

GeneralizedPolylog

The generalized polylogarithmic function

MultiPolylog

The multiple polylogarithmic function

 Calling Sequence GeneralizedPolylog($\left[{a}_{1},{a}_{2},...,{a}_{w}\right],x$) MultiPolylog($\left[{m}_{1},{m}_{2},...,{m}_{n}\right],\left[{z}_{1},{z}_{2},\mathrm{...},{z}_{n}\right]$)

Parameters

 ${a}_{1},{a}_{2},...,{a}_{w}$ - complex numbers, or algebraic expressions representing them $x$ - a complex number, or an algebraic expressions representing a such ${m}_{1},{m}_{2},...,{m}_{n}$ - positive integers ${z}_{1},{z}_{2},\mathrm{...},{z}_{n}$ - complex numbers, or algebraic expressions representing them

Description

 GeneralizedPolylog and MultiPolylog represent the function class consisting of generalized polylogarithms, multiple polylogarithms, harmonic polylogarithms, hyperlogarithms, and related functions.
 The generalized polylogarithm is defined recursively, as the iterated integral
 $\mathrm{GeneralizedPolylog}\left(\left[{a}_{i}$\mathrm{=}\left(i,1..w\right)\right],x\right)=\mathrm{%int}\left(\mathrm{/}\left(\mathrm{GeneralizedPolylog}\left(\left[{a}_{i}$\mathrm{=}\left(i,2..w\right)\right],y\right),\mathrm{-}\left(y,{a}_{1}\right)\right),\mathrm{=}\left(y,0..x\right)\right)$
 The recursion stops, as
 $\mathrm{GeneralizedPolylog}\left(\left[\right],x\right)=1$
 For all the a[i] indices being zero, an alternative definition is used, as
 $\mathrm{GeneralizedPolylog}\left(\left[0$w\right],x\right)=\frac{{\mathrm{ln}\left(x\right)}^{n}}{n!}$  The multiple polylogarithm, on the other hand, represent the sum form over  $\mathrm{MultiPolylog}\left({m}_{i}$\mathrm{=}\left(i,1..n\right),{z}_{i}$\mathrm{=}\left(i,1..n\right)\right)=\mathrm{%sum}\left(\mathrm{Multiply}\left(\mathrm{/}\left(\mathrm{^}\left({z}_{j},{i}_{j}\right),\mathrm{^}\left({i}_{j},{m}_{j}\right)\right)$\mathrm{=}\left(j,1..n\right)\right),i\right)$
 and the analytic continuation thereof outside its convergent region, which is given by the restrictions
 $\prod _{j=1}^{n}{a}_{j}$
 The relation between GeneralizedPolylog and MultiPolylog is given as
 The generalized polylogarithm and related functions show up in high energy physics, where scattering amplitudes and other observables in quantum field theories, often are given in terms of this class of functions when calculated with high precision, i.e. beyond the leading order in perturbative expansion used in the Feynman diagrammatic expansion.

Examples

 Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):
 > $\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right):$
 Functions such as ln, polylog and MultiZeta may appear as special cases of the generalized polylogarithms
 > $\mathrm{%GeneralizedPolylog}\left(\left[0\right],x\right)=\mathrm{GeneralizedPolylog}\left(\left[0\right],x\right);$
 ${\mathrm{%GeneralizedPolylog}}{}\left(\left[{0}\right]{,}{x}\right){=}{\mathrm{ln}}{}\left({x}\right)$ (1)
 > $\mathrm{%GeneralizedPolylog}\left(\left[0,0,0,0,1\right],x\right)=\mathrm{GeneralizedPolylog}\left(\left[0,0,0,0,1\right],x\right);$
 ${\mathrm{%GeneralizedPolylog}}{}\left(\left[{0}{,}{0}{,}{0}{,}{0}{,}{1}\right]{,}{x}\right){=}{-}{{\mathrm{Li}}}_{{5}}{}\left({x}\right)$ (2)

Likewise, and using a more compact input syntax

 > $\left(\mathrm{%MultiPolylog}=\mathrm{MultiPolylog}\right)\left(\left[2,3,4,5\right],\left[1,1,1,1\right]\right);$
 ${\mathrm{%MultiPolylog}}{}\left(\left[{2}{,}{3}{,}{4}{,}{5}\right]{,}\left[{1}{,}{1}{,}{1}{,}{1}\right]\right){=}{\mathrm{MultiZeta}}{}\left({2}{,}{3}{,}{4}{,}{5}\right)$ (3)

The Multiple Polylogarithm has been implemented for certain special values such as the oscillating multiple Zeta values up to weight four

 >
 ${\mathrm{%MultiPolylog}}{}\left(\left[{2}{,}{1}{,}{1}\right]{,}\left[{1}{,}{-}{1}{,}{-}{1}\right]\right){=}\frac{{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{{\mathrm{\pi }}}^{{2}}}{{8}}{-}\frac{{7}{{\mathrm{\pi }}}^{{4}}}{{288}}{+}{3}{{\mathrm{Li}}}_{{4}}{}\left(\frac{{1}}{{2}}\right){+}\frac{{{\mathrm{ln}}{}\left({2}\right)}^{{4}}}{{8}}$ (4)

and also for certain cases at weights two and three where it reduces directly to classical polylogarithms

 >
 ${\mathrm{%MultiPolylog}}{}\left(\left[{2}{,}{1}\right]{,}\left[{1}{,}{x}\right]\right){=}{{\mathrm{Li}}}_{{2}}{}\left({1}{-}{x}\right){\mathrm{ln}}{}\left({1}{-}{x}\right){-}{{\mathrm{Li}}}_{{3}}{}\left({x}\right){-}{2}{{\mathrm{Li}}}_{{3}}{}\left({1}{-}{x}\right){+}{2}{\mathrm{\zeta }}{}\left({3}\right)$ (5)

Similar relations are implemented for the generalized polylogarithm

 > $\left(\mathrm{%GeneralizedPolylog}=\mathrm{GeneralizedPolylog}\right)\left(\left[0,1,1\right],x\right)$
 ${\mathrm{%GeneralizedPolylog}}{}\left(\left[{0}{,}{1}{,}{1}\right]{,}{x}\right){=}{-}{{\mathrm{Li}}}_{{3}}{}\left({1}{-}{x}\right){+}{{\mathrm{Li}}}_{{2}}{}\left({1}{-}{x}\right){\mathrm{ln}}{}\left({1}{-}{x}\right){+}\frac{{\mathrm{ln}}{}\left({x}\right){{\mathrm{ln}}{}\left({1}{-}{x}\right)}^{{2}}}{{2}}{+}{\mathrm{\zeta }}{}\left({3}\right)$ (6)

Many relations are obeyed by the generalized polylogarithm, such as the rescaling relation

 > $\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78*I,1.99+3.33*I,0.77+0.09*I\right],1.35-1.01*I\right)$
 ${\mathrm{GeneralizedPolylog}}{}\left(\left[{0.23}{-}{1.78}{I}{,}{1.99}{+}{3.33}{I}{,}{0.77}{+}{0.09}{I}\right]{,}{1.35}{-}{1.01}{I}\right)$ (7)
 > $\mathrm{GeneralizedPolylog}\left(\left[\left(0.23-1.78I\right)z,\left(1.99+3.33I\right)z,\left(0.77+0.09I\right)z\right],\left(1.35-1.01I\right)z\right)$
 ${\mathrm{GeneralizedPolylog}}{}\left(\left[\left({0.23}{-}{1.78}{I}\right){z}{,}\left({1.99}{+}{3.33}{I}\right){z}{,}\left({0.77}{+}{0.09}{I}\right){z}\right]{,}\left({1.35}{-}{1.01}{I}\right){z}\right)$ (8)

Evaluate numerically (7) and (8) up to 8 digits

 >
 ${0.013040566}{+}{0.21053300}{I}{=}{0.013040566}{+}{0.21053300}{I}$ (9)

and the shuffle relation

 > $\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78I\right],1.35-1.01I\right)\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33I,0.77+0.09I\right],1.35-1.01I\right)$
 $\left({-0.2780299456}{-}{1.097010462}{I}\right){\mathrm{GeneralizedPolylog}}{}\left(\left[{1.99}{+}{3.33}{I}{,}{0.77}{+}{0.09}{I}\right]{,}{1.35}{-}{1.01}{I}\right)$ (10)
 > $\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78I,1.99+3.33I,0.77+0.09I\right],1.35-1.01I\right)+\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33I,0.23-1.78I,0.77+0.09I\right],1.35-1.01I\right)+\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33I,0.77+0.09I,0.23-1.78I\right],1.35-1.01I\right)$
 ${\mathrm{GeneralizedPolylog}}{}\left(\left[{0.23}{-}{1.78}{I}{,}{1.99}{+}{3.33}{I}{,}{0.77}{+}{0.09}{I}\right]{,}{1.35}{-}{1.01}{I}\right){+}{\mathrm{GeneralizedPolylog}}{}\left(\left[{1.99}{+}{3.33}{I}{,}{0.23}{-}{1.78}{I}{,}{0.77}{+}{0.09}{I}\right]{,}{1.35}{-}{1.01}{I}\right){+}{\mathrm{GeneralizedPolylog}}{}\left(\left[{1.99}{+}{3.33}{I}{,}{0.77}{+}{0.09}{I}{,}{0.23}{-}{1.78}{I}\right]{,}{1.35}{-}{1.01}{I}\right)$ (11)

Up to 6 digits,

 > $\mathrm{evalf}\left[6\right]\left(=\right)$
 ${0.264849}{+}{0.438022}{I}{=}{0.264849}{+}{0.438022}{I}$ (12)

and the "stuffle" relation

 > $\mathrm{%MultiPolylog}\left(\left[2\right],\left[0.98-0.11I\right]\right)\mathrm{%MultiPolylog}\left(\left[3\right],\left[2.77-1.04I\right]\right)$
 ${\mathrm{%MultiPolylog}}{}\left(\left[{2}\right]{,}\left[{0.98}{-}{0.11}{I}\right]\right){\mathrm{%MultiPolylog}}{}\left(\left[{3}\right]{,}\left[{2.77}{-}{1.04}{I}\right]\right)$ (13)
 > $\mathrm{%MultiPolylog}\left(\left[2,3\right],\left[0.98-0.11I,2.77-1.04I\right]\right)+\mathrm{%MultiPolylog}\left(\left[3,2\right],\left[2.77-1.04I,0.98-0.11I\right]\right)+\mathrm{%MultiPolylog}\left(\left[5\right],\left[\left(0.98-0.11I\right)\left(2.77-1.04I\right)\right]\right)$
 ${\mathrm{%MultiPolylog}}{}\left(\left[{2}{,}{3}\right]{,}\left[{0.98}{-}{0.11}{I}{,}{2.77}{-}{1.04}{I}\right]\right){+}{\mathrm{%MultiPolylog}}{}\left(\left[{3}{,}{2}\right]{,}\left[{2.77}{-}{1.04}{I}{,}{0.98}{-}{0.11}{I}\right]\right){+}{\mathrm{%MultiPolylog}}{}\left(\left[{5}\right]{,}\left[{2.6002}{-}{1.3239}{I}\right]\right)$ (14)
 > $\mathrm{evalf}\left[4\right]\left(\mathrm{value}\left(=\right)\right)$
 ${2.809}{-}{4.448}{I}{=}{2.809}{-}{4.448}{I}$ (15)

References

 [1] A.B.Goncharov. "Multiple polylogarithms, cyclotomy and modular complexes", Math Res.Letters. Vol. 5 (1998): 497-516. [2] Jens Vollinga, Stefan Weinzierl.  "Numerical evaluation of multiple polylogarithms", Comput.Phys.Commun. Vol. 167 (2005): 23 pp. [3] H. Frellesvig, D. Tommasini, C. Wever.  "On the reduction of generalized polylogarithms to Li_n and Li_22 and on the evaluation thereof", JHEP 1603 (2016): 35pp

Compatibility

 • The GeneralizedPolylog command was introduced in Maple 2018.
 • The MultiPolylog command was introduced in Maple 2018.