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 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)

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

• Andrs 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.