Mathematical Functions

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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

$GeneralizedPolylog ([a_{1}, ..., a_{w}], x) = \int_{0}^{x} \frac{GeneralizedPolylog ([a_{2}, ..., a_{w}], y)}{y - a_{1}} &DifferentialD; y$

The recursion stops, as

$GeneralizedPolylog ([], x) = 1$

For all the $a_{i}$ indices being zero, an alternative definition is used, as

$GeneralizedPolylog ([\underset{w times}{\underset{⏟}{0, ..., 0}}], x) = \frac{{\ln (x)}^{n}}{n!}$

The multiple polylogarithm, on the other hand, represent the sum form over $i_{1} > i_{2} > .. &period; > i__n > 0$

$MultiPolylog (m_{1}, ..., m_{n}, z_{1}, ..., z_{n}) = \sum_{i} \prod_{}^{} (\frac{z_{1}^{i_{1}}}{i_{1}^{m_{1}}}, ..., \frac{z_{n}^{i_{n}}}{i_{n}^{m_{n}}})$

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

$\prod_{j = 1}^{n} a_{j}$

$|z_{1}| < 1, |z_{1} z_{2}| < 1, \dots, |\prod_{i = 1}^{n} z_{i}| < 1$

MultiZeta is an implementation of multiple zeta values, also known as the generalized Euler sums over $i_{1} > i_{2} > .. &period; > i__n > 0$

$MultiZeta (m_{1}, ..., m_{n}) = \sum_{i} \prod_{}^{} (\frac{1}{i_{1}^{m_{1}}}, ..., \frac{1}{i_{n}^{m_{n}}})$

The sum converges for all positive integer arguments, except when the first argument equals one, for instance as in $MultiZeta (1, 2, 3)$ , in which case the function diverges.

Examples

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

>	$interface (typesetting = extended) &colon; Typesetting :- EnableTypesetRule (Typesetting :- SpecialFunctionRules) &colon;$

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

>	$%GeneralizedPolylog ([0], x) = GeneralizedPolylog ([0], x) &semi;$

$GeneralizedPolylog ([0], x) = \ln (x)$

(1)

>	$%GeneralizedPolylog ([0, 0, 0, 0, 1], x) = GeneralizedPolylog ([0, 0, 0, 0, 1], x) &semi;$

$GeneralizedPolylog ([0, 0, 0, 0, 1], x) = - {Li}_{5} (x)$

(2)

Likewise, and using a more compact input syntax

>	$(%MultiPolylog = MultiPolylog) ([2, 3, 4, 5], [1, 1, 1, 1]) &semi;$

$MultiPolylog ([2, 3, 4, 5], [1, 1, 1, 1]) = MultiZeta (2, 3, 4, 5)$

(3)

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

>	$(%MultiPolylog = MultiPolylog) ([2, 1, 1], [1, - 1, - 1])$

$MultiPolylog ([2, 1, 1], [1, −1, −1]) = \frac{{\ln (2)}^{2} π^{2}}{8} - \frac{7 π^{4}}{288} + 3 {Li}_{4} (\frac{1}{2}) + \frac{{\ln (2)}^{4}}{8}$

(4)

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

>	$(%MultiPolylog = MultiPolylog) ([2, 1], [1, x])$

$MultiPolylog ([2, 1], [1, x]) = {Li}_{2} (1 - x) \ln (1 - x) - {Li}_{3} (x) - 2 {Li}_{3} (1 - x) + 2 ζ (3)$

(5)

Similar relations are implemented for the generalized polylogarithm

>	$(%GeneralizedPolylog = GeneralizedPolylog) ([0, 1, 1], x)$

$GeneralizedPolylog ([0, 1, 1], x) = - {Li}_{3} (1 - x) + {Li}_{2} (1 - x) \ln (1 - x) + \frac{\ln (x) {\ln (1 - x)}^{2}}{2} + ζ (3)$

(6)

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

>	$GeneralizedPolylog ([0.23 - 1.78 &ast; I, 1.99 + 3.33 &ast; I, 0.77 + 0.09 &ast; I], 1.35 - 1.01 &ast; I)$

$GeneralizedPolylog ([0.23 - 1.78 I, 1.99 + 3.33 I, 0.77 + 0.09 I], 1.35 - 1.01 I)$

(7)

>	$GeneralizedPolylog ([(0.23 - 1.78 I) z, (1.99 + 3.33 I) z, (0.77 + 0.09 I) z], (1.35 - 1.01 I) z)$

$GeneralizedPolylog ([(0.23 - 1.78 I) z, (1.99 + 3.33 I) z, (0.77 + 0.09 I) z], (1.35 - 1.01 I) z)$

(8)

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

>	$evalf [8] (eval (=, [z = 1.91 - 0.39 I]))$

$0.013040566 + 0.21053300 I = 0.013040566 + 0.21053300 I$

(9)

and the shuffle relation

>	$GeneralizedPolylog ([0.23 - 1.78 I], 1.35 - 1.01 I) GeneralizedPolylog ([1.99 + 3.33 I, 0.77 + 0.09 I], 1.35 - 1.01 I)$

$(−0.2780299456 - 1.097010462 I) GeneralizedPolylog ([1.99 + 3.33 I, 0.77 + 0.09 I], 1.35 - 1.01 I)$

(10)

>	$GeneralizedPolylog ([0.23 - 1.78 I, 1.99 + 3.33 I, 0.77 + 0.09 I], 1.35 - 1.01 I) + GeneralizedPolylog ([1.99 + 3.33 I, 0.23 - 1.78 I, 0.77 + 0.09 I], 1.35 - 1.01 I) + GeneralizedPolylog ([1.99 + 3.33 I, 0.77 + 0.09 I, 0.23 - 1.78 I], 1.35 - 1.01 I)$

$GeneralizedPolylog ([0.23 - 1.78 I, 1.99 + 3.33 I, 0.77 + 0.09 I], 1.35 - 1.01 I) + GeneralizedPolylog ([1.99 + 3.33 I, 0.23 - 1.78 I, 0.77 + 0.09 I], 1.35 - 1.01 I) + GeneralizedPolylog ([1.99 + 3.33 I, 0.77 + 0.09 I, 0.23 - 1.78 I], 1.35 - 1.01 I)$

(11)

Up to 6 digits,

>	$evalf [6] (=)$

$0.264849 + 0.438022 I = 0.264849 + 0.438022 I$

(12)

and the "stuffle" relation

>	$%MultiPolylog ([2], [0.98 - 0.11 I]) %MultiPolylog ([3], [2.77 - 1.04 I])$

$MultiPolylog ([2], [0.98 - 0.11 I]) MultiPolylog ([3], [2.77 - 1.04 I])$

(13)

>	$%MultiPolylog ([2, 3], [0.98 - 0.11 I, 2.77 - 1.04 I]) + %MultiPolylog ([3, 2], [2.77 - 1.04 I, 0.98 - 0.11 I]) + %MultiPolylog ([5], [(0.98 - 0.11 I) (2.77 - 1.04 I)])$

$MultiPolylog ([2, 3], [0.98 - 0.11 I, 2.77 - 1.04 I]) + MultiPolylog ([3, 2], [2.77 - 1.04 I, 0.98 - 0.11 I]) + MultiPolylog ([5], [2.6002 - 1.3239 I])$

(14)

>	$evalf [4] (value (=))$

$2.809 - 4.448 I = 2.809 - 4.448 I$

(15)

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

>	$%MultiZeta (43) = MultiZeta (43)$

$MultiZeta (43) = ζ (43)$

(16)

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

>	$(%MultiZeta = MultiZeta) (27, 27)$

$MultiZeta (27, 27) = \frac{{ζ (27)}^{2}}{2} - \frac{ζ (54)}{2}$

(17)

and of two arguments summing to an odd number

>	$(%MultiZeta = MultiZeta) (11, 8) &semi;$

$MultiZeta (11, 8) = - \frac{75583 ζ (19)}{2} + \frac{9724 π^{2} ζ (17)}{3} + \frac{4433 π^{4} ζ (15)}{90} + \frac{286 π^{6} ζ (13)}{315} + \frac{121 π^{8} ζ (11)}{9450} + \frac{8 π^{10} ζ (9)}{93555}$

(18)

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

>	$(%MultiZeta = MultiZeta) (2, 1, 4)$

$MultiZeta (2, 1, 4) = \frac{7 π^{4} ζ (3)}{360} - \frac{11 π^{2} ζ (5)}{12} + \frac{61 ζ (7)}{8}$

(19)

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

>	$(%MultiPolylog = MultiPolylog) ([2, 3, 4, 5], [1, 1, 1, 1]) &semi;$

$MultiPolylog ([2, 3, 4, 5], [1, 1, 1, 1]) = MultiZeta (2, 3, 4, 5)$

(20)

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

>	$MultiZeta (7, 9) MultiZeta (6)$

$\frac{MultiZeta (7, 9) π^{6}}{945}$

(21)

>	$MultiZeta (7, 9, 6) + MultiZeta (7, 6, 9) + MultiZeta (6, 7, 9) + MultiZeta (13, 9) + MultiZeta (7, 15)$

$MultiZeta (7, 9, 6) + MultiZeta (7, 6, 9) + MultiZeta (6, 7, 9) + MultiZeta (13, 9) + MultiZeta (7, 15)$

(22)

Up to 5 digits,

>	$evalf [5] (=)$

$0.0084952 = 0.0084952$

(23)

and the duality

>	$MultiZeta (2, 3, 4)$

$MultiZeta (2, 3, 4)$

(24)

>	$MultiZeta (2, 1, 1, 2, 1, 2)$

$MultiZeta (2, 1, 1, 2, 1, 2)$

(25)

>	$evalf (=)$

$0.06781184623 = 0.06781184623$

(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.

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