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Harmonic Polylogarithm

The harmonic polylogarithm H(a_1,...,a_k;z) is an iterated integral with indices a_j in {-1,0,1}. It can be expressed using the generalized polylogarithm as

H(a_1,...,a_k;z)=(-1)^rG(a_1,...,a_k;z),

where r is the number of indices equal to 1. Equivalently, its integration kernels for indices -1, 0, and 1 are 1/(1+t), 1/t, and 1/(1-t), respectively. For 0<z<1, the one-index cases are H(-1;z)=ln(1+z), H(0;z)=lnz, and H(1;z)=-ln(1-z) (Remiddi and Vermaseren 2000).

The harmonic polylogarithm is implemented in the Wolfram Language as HarmonicPolyLog[{a1, ..., ak}, z].

See also

Generalized Polylogarithm, Iterated Integral, Nielsen Generalized Polylogarithm

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References

Remiddi, E. and Vermaseren, J. A. M. "Harmonic Polylogarithms." Int. J. Mod. Phys. A 15, 725-754, 2000. https://doi.org/10.1142/S0217751X00000367.

Cite this as:

Weisstein, Eric W. "Harmonic Polylogarithm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicPolylogarithm.html

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