PolyLog.jl

The PolyLog.jl package provides Julia implementations of real and complex polylogarithms, including the real and complex dilogarithm and trilogarithm.

Example

using PolyLog
# real polylogarithms for real argumentsreli1(1.0) # Re[Li_1(x)]reli2(1.0) # Re[Li_2(x)] (dilogarithm)reli3(1.0) # Re[Li_3(x)] (trilogarithm)reli4(1.0) # Re[Li_4(x)]reli(10, 1.0) # Re[Li_n(x)] for all integers n (here: n = 10)reli(10, big"1.0") # Re[Li_n(x)] for all integers n (here: n = 10)reli(-2, 1.0) # Re[Li_n(x)] for all integers n (here: n = -2)# complex polylogarithms for real or complex argumentsli0(1.0+1.0im) # Li_0(z)li1(1.0+1.0im) # Li_1(z)li2(1.0+1.0im) # Li_2(z) (dilogarithm)li3(1.0+1.0im) # Li_3(z) (trilogarithm)li4(1.0+1.0im) # Li_4(z)li5(1.0+1.0im) # Li_5(z)li6(1.0+1.0im) # Li_6(z)li(10, 1.0+1.0im) # Li_n(z) for all integers n (here: n = 10)li(10, big"1.0"+1im) # Li_n(z) for all integers n (here: n = 10)li(-2, 1.0+1.0im) # Li_n(z) for all integers n (here: n = -2)

Example using ForwardDiff

using ForwardDiff, PolyLog
ForwardDiff.derivative(reli1, 0.5) # Re[Li_1]'(x)
ForwardDiff.derivative(reli2, 0.5) # Re[Li_2]'(x)
ForwardDiff.derivative(reli3, 0.5) # Re[Li_3]'(x)
ForwardDiff.derivative(reli4, 0.5) # Re[Li_4]'(x)
ForwardDiff.derivative(x ->reli(10, x), 0.5) # Re[Li_n]'(x) for n = 10
ForwardDiff.derivative(x ->reli(10, x), big"0.5") # Re[Li_n]'(x) for n = 10
ForwardDiff.derivative(x ->reli(-2, x), 0.5) # Re[Li_n]'(x) for n = -2

Documentation

https://docs.juliahub.com/PolyLog/

Notes

The implementation of the real dilogarithm is an adaptation of [arXiv:2201.01678].

The implementation of the complex dilogarithm has been inspired by the implementation in SPheno and has been translated to Julia.

The implementation of the real trilogarithm is an adaptation of [arXiv:2308.11619].

The implementation of the general n-th order polylogarithm is an adaptation of [arXiv:2010.09860].

Contributors

• Andrés Riedemann (@longemen3000): Extensions for ForwardDiff.jl and ChainRules.jl.

Copying

PolyLog.jl is licenced under the MIT License.

Links

Refer to the package Polylogarithms.jl for a Julia implementation of polylogarithms of arbitrary complex order.