The PolyLog.jl package provides Julia implementations of real and complex polylogarithms, including the real and complex dilogarithm and trilogarithm.
using PolyLog
# real polylogarithms for real arguments
reli1(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 arguments
li0(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)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 = -2https://docs.juliahub.com/PolyLog/
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].
- Andrés Riedemann (@longemen3000): Extensions for ForwardDiff.jl and ChainRules.jl.
PolyLog.jl is licenced under the MIT License.
Refer to the package Polylogarithms.jl for a Julia implementation of polylogarithms of arbitrary complex order.