Polylogarithms

This implements the Polylogarithm and some related functions that were needed (Harmonic numbers, Stieltjes constants, and Bernoulli numbers and polynomials, and Euler numbers).

The code is aimed at calculating Li_s(z) for all (complex) s and z.

This is still a little experimental, but there is a fairly large test set that all works nicely.

Note that the aimed for accuracy is 1.0e-12 relative error, but that occasional errors as large as 1.0e-11 have been seen.

Functions

• polylog(s, z) the polylogarithm function

• bernoulli(n) Provides the first 59 Bernoulli numbers as exact rationals

• bernoulli(n,x) Provides the Bernoulli polynomials

• euler(n) Provides the first 61 Euler numbers as BigInt

• harmonic(n) Provides the Harmonic numbers

• harmonic(n,r) Provides the generalised Harmonic numbers

• stieltjes(n) Provides the first 10 Stieltjes constants (see Abramowitz and Stegun, 23.2.5), also known as the generalized Euler-Mascheroni constants.

• dirichlet_beta(z) Provides the Dirichlet beta function (the eta function is included in SpecialFunctions)

• rogers(z) Rogers L-function

• spence(z) and dilog -- aliases for polylog(2, z)

• trilog(z) and tetralog -- aliaes for polylog(3, z) and polylog(3, z)

Recent additions

Currently in the process of adding functions to calculate the derivatives of the polylogarithm WRT to both z and s. Doing this for a wide range of arguments requires derivatives of the Riemann zeta function so this is also in the process of being added.

Notes

There are few extra functions exported that are used as components here. These are currently being used in testing, but ultimately will taken out of the export list, so don't use these unless you are clear about what you are doing.

Examples

julia> using Polylogarithms
julia> polylog(2.0, 1.0)
1.6449340668482273

Details and Accuracy

Extended details of the algorithms being used here are provided at https://arxiv.org/abs/2010.09860. However note that this reflected performance of v0.1, which has been substantially improved for v0.2.

Relative accuracy has been tested to be within 1.0e-12 over a wide range of inputs for

• s in the 16x16 square in the complex plane bounded by +- 8
• z in the square in the complex plane bounded by +- 1.0e20

It fails to achieve this accuracy in <<1% of cases.

Many of the tests are in the tests directory, but the larger body of tests, and plots visualising the results are the extended_tests directory using the precomputed reference data in the data directory. Running presumes you can write files (eg plots)

Note

According to naming conventions, this package should have been called Polylogarithm, but there is already an older package doing polylogarithms, https://github.com/Expander/polylogarithm but it's using C/CPP/Fortran bindings, and only appears to do s=2,3,4,5,6.

There is a new package PolyLog, which does some newer stuff for dilogarithms, so that might be for you if you only care about that case.

Changes

Changes from v0.1 to v0.2 are in Changes.txt, but most of the changes are improvements. You should see few changes to any interfaces unless you are using the diagnostic output or using non-exported functions.