{"id":32157651,"url":"https://github.com/mroughan/polylogarithms.jl","last_synced_at":"2025-10-21T12:51:58.927Z","repository":{"id":41998822,"uuid":"284601903","full_name":"mroughan/Polylogarithms.jl","owner":"mroughan","description":"Polylogarithm function and related special functions and 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Polylogarithms\n\n[![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://mroughan.github.io/Polylogarithms.jl/stable)\n\u003c!-- [![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://matthew.roughan@adelaide.edu.au.github.io/Polylogarithms.jl/dev) --\u003e\n[![Build Status](https://travis-ci.com/mroughan/Polylogarithms.jl.svg?branch=master)](https://travis-ci.com/mroughan/Polylogarithms.jl)\n[![Coverage](https://codecov.io/gh/mroughan/Polylogarithms.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/mroughan/Polylogarithms.jl)\n\nThis implements the\n[Polylogarithm](https://en.wikipedia.org/wiki/Polylogarithm#Relationship_to_other_functions)\nand some related functions that were needed (Harmonic numbers,\nStieltjes constants, and Bernoulli numbers and polynomials, and Euler numbers).\n\nThe code is aimed at calculating Li_s(z) for all (complex) s and z. \n\nThis is still a little experimental, but there is a fairly large test\nset that all works nicely.\n\nNote that the aimed for accuracy is 1.0e-12 relative error, but that\noccasional errors as large as 1.0e-11 have been seen. \n\n# Functions\n\n + `polylog(s, z)` the polylogarithm function\n \n + `bernoulli(n)`  Provides the first 59 Bernoulli numbers as exact rationals\n + `bernoulli(n,x)`  Provides the Bernoulli polynomials\n \n + `euler(n)`  Provides the first 61 Euler numbers as BigInt \n\n + `harmonic(n)` Provides the Harmonic numbers\n + `harmonic(n,r)` Provides the generalised Harmonic numbers\n \n + `stieltjes(n)` Provides the first 10 [Stieltjes\n constants](https://en.wikipedia.org/wiki/Stieltjes_constants) (see\n Abramowitz and Stegun, 23.2.5), also known as the generalized\n Euler-Mascheroni constants.\n \n + `dirichlet_beta(z)` Provides the Dirichlet beta function\n \n\n# Examples\n\n```\njulia\u003e using Polylogarithms\njulia\u003e polylog(2.0, 1.0)\n1.6449340668482273\n```\n\n# Details and Accuracy\n\nExtended details of the algorithms being used here are provided at\nhttps://arxiv.org/abs/2010.09860.\n\nAccuracy has been tested over a wide range of scenarios. It starts to\nfalter for large $z$ and $\\Re(s) \u003e 8$. \n\n# Note\n\nAccording to naming conventions, this package should have been called\nPolylogarithm, but there is already an older package doing\npolylogarithms, https://github.com/Expander/polylogarithm but it's\nusing C/CPP/Fortran bindings, and only appears to do s=2,3,4,5,6.\n\nThere is a new package\n[PolyLog](https://github.com/Expander/PolyLog.jl), which does some\nnewer stuff for dilogarithms, so that might be for you if you care\nabout that case. \n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmroughan%2Fpolylogarithms.jl","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fmroughan%2Fpolylogarithms.jl","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmroughan%2Fpolylogarithms.jl/lists"}