implement polylog for arbitrarily large arguments.

We’ve been running into an issue over at PlasmaPy, where when we try to use polylog to calculate the Fermi Integral we hit an implementation error:

import numpy as np
import mpmath
mpmath.polylog(1 + 0.5, -np.exp(3.89))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/user/anaconda3/lib/python3.6/site-packages/mpmath/ctx_mp_python.py", line 1016, in f_wrapped
    retval = f(ctx, *args, **kwargs)
  File "/home/user/anaconda3/lib/python3.6/site-packages/mpmath/functions/zeta.py", line 482, in polylog
    return polylog_general(ctx, s, z)
  File "/home/user/anaconda3/lib/python3.6/site-packages/mpmath/functions/zeta.py", line 449, in polylog_general
    raise NotImplementedError("polylog for arbitrary s and z")
NotImplementedError: polylog for arbitrary s and z

Would it be possible to implement polylog for large arguments? For our purposes it is fine to restrict it to an index of 1.5 for now.

[grozin]

Can this help? https://arxiv.org/abs/1809.07084 Ordinary polylogarithms are a particular case of harmonic polylogarithms. And this paper provides a method to reduce (analytically) polylogarithms of any real argument to combinations of polylogarithms of arguments between 0 and $\sqrt{2}-1$. At least up to weight 8.

[fredrik-johansson]

Yes, it would be possible. The decomposition in terms of the Hurwitz zeta function could be used for this. Here is code that should work at least in your case:

def polylogb(s,z):
    s = mpmathify(s)
    z = mpmathify(z)
    v = 1-s
    y = log(-z)/(2*pi*j)
    return gamma(v)*(1j**v*zeta(v,0.5+y) + 1j**-v*zeta(v,0.5-y))/(2*pi)**v

>>> polylogb(1+0.5, -exp(3.89))
mpc(real='-6.2729922293118188', imag='0.0')

Beware that there could be some ranges where this formula loses precision.