https://github.com/mroughan/polylogarithms.jl

Polylogarithm function and related special functions and sequences.
https://github.com/mroughan/polylogarithms.jl

julia math special-functions

Last synced: 8 months ago
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Polylogarithm function and related special functions and sequences.

• Host: GitHub
• URL: https://github.com/mroughan/polylogarithms.jl
• Owner: mroughan
• License: mit
• Created: 2020-08-03T04:20:35.000Z (almost 6 years ago)
• Default Branch: master
• Last Pushed: 2023-11-21T16:13:33.000Z (over 2 years ago)
• Last Synced: 2025-10-21T12:51:51.841Z (8 months ago)
• Topics: julia, math, special-functions
• Language: Julia
• Homepage:
• Size: 6.37 MB
• Stars: 10
• Watchers: 1
• Forks: 7
• Open Issues: 14
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
# Polylogarithms

[![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://mroughan.github.io/Polylogarithms.jl/stable)

[![Build Status](https://travis-ci.com/mroughan/Polylogarithms.jl.svg?branch=master)](https://travis-ci.com/mroughan/Polylogarithms.jl)

[![Coverage](https://codecov.io/gh/mroughan/Polylogarithms.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/mroughan/Polylogarithms.jl)

This implements the

[Polylogarithm](https://en.wikipedia.org/wiki/Polylogarithm#Relationship_to_other_functions)

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](https://en.wikipedia.org/wiki/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

 

# 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.

Accuracy has been tested over a wide range of scenarios. It starts to

falter for large $z$ and $\Re(s) > 8$. 

# 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](https://github.com/Expander/PolyLog.jl), which does some

newer stuff for dilogarithms, so that might be for you if you care

about that case.