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Mathematical Functions

Relevant developments in the MathematicalFunctions project happened for Maple 2018, with the addition to the Maple library of the GeneralizedPolylog, MultiPolylog and MultiZeta functions.

Generalized polylogarithms

 

  

Generalized polylogarithms [1, 2] (also known as Goncharov polylogarithms, generalized harmonic polylogarithms, or hyperlogarithms) are a class of functions that frequently show up in results for Feynman integrals, as they appear in high energy physics (for overview articles, see e.g. refs. [3, 4]). Therefore tools for their manipulation and evaluation are of high importance for precise predictions in high energy particle scattering processes, as they take place for instance at the Large Hadron Collider at CERN.

Generalized polylogarithms are also a generalization of functions such as the logarithm, the classical (or Euler) polylogarithm, and the harmonic polylogarithm [5], which all appear as special cases.

When evaluated at certain special values, generalized polylogarithms reduce to a set of numbers called multiple zeta values [6, 7, 8], which are a generalization of the values of the Riemann zeta function evaluated at positive integers. Aside form their appearance in physics, these numbers are also of interest in pure mathematics such as number theory.

  

 

  

The generalized polylogarithm is defined recursively, as the iterated integral

  

GeneralizedPolylogai$`=`i,1..w,x=%int`/`GeneralizedPolylogai$`=`i,2..w,y,`-`y,a1,`=`y,0..x

  

The recursion stops, as

  

GeneralizedPolylog,x=1

  

For all the ai indices being zero, an alternative definition is used, as

  

GeneralizedPolylog0$w,x=lnxnn!

  

 

  

The multiple polylogarithm, on the other hand, represent the sum form over i1>i2>...>i__n >0

  

MultiPolylogmi$`=`i,1..n,zi$`=`i,1..n=%sumMultiply`/``^`zj,ij,`^`ij,mj$`=`j,1..n,i

  

and the analytic continuation thereof outside its convergent region, which is given by the restrictions

  

j=1naj

  

z1<1,z1z2<1,    &comma;%productzi&comma;`=`i&comma;1..n<1

 

MultiZeta is an implementation of multiple zeta values, also known as the generalized Euler sums over i1&gt;i2&gt;..&period;&gt;i__n &gt;0

MultiZetami&dollar;`=`i&comma;1..n=%sumMultiply`/`1&comma;`^`ij&comma;mj&dollar;`=`j&comma;1..n&comma;i

 

  

The sum converges for all positive integer arguments, except when the first argument equals one, for instance as in MultiZeta1&comma;2&comma;3, in which case the function diverges.

Examples

 

To display special functions using textbook notation, use extended typesetting and enable the typesetting of mathematical functions.

 

interfacetypesetting &equals; extended&colon; Typesetting:-EnableTypesetRuleTypesetting:-SpecialFunctionRules&colon;

  

Functions such as ln, polylog and MultiZeta may appear as special cases of the generalized polylogarithms

%GeneralizedPolylog0&comma;x&equals;GeneralizedPolylog0&comma;x&semi;

%GeneralizedPolylog0&comma;x=lnx

(1)

%GeneralizedPolylog0&comma;0&comma;0&comma;0&comma;1&comma;x&equals;GeneralizedPolylog0&comma;0&comma;0&comma;0&comma;1&comma;x&semi;

%GeneralizedPolylog0&comma;0&comma;0&comma;0&comma;1&comma;x=Li5x

(2)

Likewise, and using a more compact input syntax

%MultiPolylog&equals;MultiPolylog2&comma;3&comma;4&comma;5&comma;1&comma;1&comma;1&comma;1&semi;

%MultiPolylog2&comma;3&comma;4&comma;5&comma;1&comma;1&comma;1&comma;1=MultiZeta2&comma;3&comma;4&comma;5

(3)

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

%MultiPolylog &equals; MultiPolylog2&comma;1&comma;1&comma;1&comma;1&comma;1

%MultiPolylog2&comma;1&comma;1&comma;1&comma;1&comma;1=ln22π287π4288+3Li412+ln248

(4)

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

%MultiPolylog &equals; MultiPolylog2&comma;1&comma;1&comma;x

%MultiPolylog2&comma;1&comma;1&comma;x=Li21xln1xLi3x2Li31x+2ζ3

(5)

Similar relations are implemented for the generalized polylogarithm

%GeneralizedPolylog&equals;GeneralizedPolylog0&comma;1&comma;1&comma;x

%GeneralizedPolylog0&comma;1&comma;1&comma;x=Li31x+Li21xln1x+lnxln1x22+ζ3

(6)

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

GeneralizedPolylog0.231.78&ast;I&comma;1.99&plus;3.33&ast;I&comma;0.77&plus;0.09&ast;I&comma;1.351.01&ast;I

GeneralizedPolylog0.231.78I&comma;1.99+3.33I&comma;0.77+0.09I&comma;1.351.01I

(7)

GeneralizedPolylog0.231.78Iz&comma;1.99+3.33Iz&comma;0.77+0.09Iz&comma;1.351.01Iz

GeneralizedPolylog0.231.78Iz&comma;1.99+3.33Iz&comma;0.77+0.09Iz&comma;1.351.01Iz

(8)

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

evalf8eval&equals;&comma;z&equals;1.910.39  I

0.013040566+0.21053300I=0.013040566+0.21053300I

(9)

and the shuffle relation

GeneralizedPolylog0.231.78I&comma;1.351.01IGeneralizedPolylog1.99+3.33I&comma;0.77+0.09I&comma;1.351.01I

−0.27802994561.097010462IGeneralizedPolylog1.99+3.33I&comma;0.77+0.09I&comma;1.351.01I

(10)

GeneralizedPolylog0.231.78I&comma;1.99+3.33I&comma;0.77+0.09I&comma;1.351.01I+GeneralizedPolylog1.99+3.33I&comma;0.231.78I&comma;0.77+0.09I&comma;1.351.01I+GeneralizedPolylog1.99+3.33I&comma;0.77+0.09I&comma;0.231.78I&comma;1.351.01I

GeneralizedPolylog0.231.78I&comma;1.99+3.33I&comma;0.77+0.09I&comma;1.351.01I+GeneralizedPolylog1.99+3.33I&comma;0.231.78I&comma;0.77+0.09I&comma;1.351.01I+GeneralizedPolylog1.99+3.33I&comma;0.77+0.09I&comma;0.231.78I&comma;1.351.01I

(11)

Up to 6 digits,

evalf6&equals;

0.264849+0.438022I=0.264849+0.438022I

(12)

and the "stuffle" relation

%MultiPolylog2&comma;0.980.11I%MultiPolylog3&comma;2.771.04I

%MultiPolylog2&comma;0.980.11I%MultiPolylog3&comma;2.771.04I

(13)

%MultiPolylog2&comma;3&comma;0.980.11I&comma;2.771.04I+%MultiPolylog3&comma;2&comma;2.771.04I&comma;0.980.11I+%MultiPolylog5&comma;0.980.11I2.771.04I

%MultiPolylog2&comma;3&comma;0.980.11I&comma;2.771.04I+%MultiPolylog3&comma;2&comma;2.771.04I&comma;0.980.11I+%MultiPolylog5&comma;2.60021.3239I

(14)

evalf4value&equals;

2.8094.448I=2.8094.448I

(15)
  

For one argument, MultiZeta reduces to the Riemann Zeta function:

%MultiZeta43&equals;MultiZeta43

%MultiZeta43=ζ43

(16)

The more relevant special cases are computed automatically, such as that of two identical arguments, here using a more compact input syntax

%MultiZeta&equals;MultiZeta27&comma;27

%MultiZeta27&comma;27=ζ2722ζ542

(17)

and of two arguments summing to an odd number

%MultiZeta&equals;MultiZeta11&comma;8&semi;

%MultiZeta11&comma;8=75583ζ192+9724π2ζ173+4433π4ζ1590+286π6ζ13315+121π8ζ119450+8π10ζ993555

(18)

All Multiple Zeta values of weight less than or equal to seven, can be written solely in terms of classical Zeta values:

%MultiZeta&equals;MultiZeta2&comma;1&comma;4

%MultiZeta2&comma;1&comma;4=7π4ζ336011π2ζ512+61ζ78

(19)

The multiple Zeta values are a special case of the the multiple polylogarithm:

%MultiPolylog&equals;MultiPolylog2&comma;3&comma;4&comma;5&comma;1&comma;1&comma;1&comma;1&semi;

%MultiPolylog2&comma;3&comma;4&comma;5&comma;1&comma;1&comma;1&comma;1=MultiZeta2&comma;3&comma;4&comma;5

(20)

The multiple zeta values obey a large number of identities, primarily the stuffle relation:

MultiZeta7&comma;9MultiZeta6

MultiZeta7&comma;9π6945

(21)

MultiZeta7&comma;9&comma;6+MultiZeta7&comma;6&comma;9+MultiZeta6&comma;7&comma;9+MultiZeta13&comma;9+MultiZeta7&comma;15

MultiZeta7&comma;9&comma;6+MultiZeta7&comma;6&comma;9+MultiZeta6&comma;7&comma;9+MultiZeta13&comma;9+MultiZeta7&comma;15

(22)

Up to 5 digits,

evalf5&equals;

0.0084952=0.0084952

(23)

and the duality

MultiZeta2&comma;3&comma;4

MultiZeta2&comma;3&comma;4

(24)

MultiZeta2&comma;1&comma;1&comma;2&comma;1&comma;2

MultiZeta2&comma;1&comma;1&comma;2&comma;1&comma;2

(25)

evalf&equals;

0.06781184623=0.06781184623

(26)

References

  

[1] A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math.Res.Lett. 5 (1998) 497 516. arXiv:1105.2076, doi:10.4310/MRL.1998.v5.n4.a7.

  

[2] A. B. Goncharov, M. Spradlin, C. Vergu, A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605. arXiv:1006.5703, doi:10.1103/PhysRevLett.105.151605.

  

[3] J. M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A48 (2015) 153001. arXiv:1412.2296, doi:10.1088/1751- 8113/48/15/153001.

  

[4] C. Duhr, Mathematical aspects of scattering amplitudes, in: Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders (TASI 2014) Boulder, Colorado, June 2-27, 2014, 2014. arXiv:1411.7538.

See Also

MathematicalFunctions

FunctionAdvisor

GeneralizedPolylog

MultiPolylog

MultiZeta

polylog