"""Weight 5 harmonic sums."""
import numba as nb
from eko.constants import zeta5
from .polygamma import cern_polygamma
[docs]
@nb.njit(cache=True)
def S5(N):
r"""Compute the harmonic sum :math:`S_5(N)`.
.. math::
S_5(N) = \sum\limits_{j=1}^N \frac 1 {j^5} = \frac 1 24 \psi_4(N+1)+\zeta(5)
with :math:`\psi_4(N)` the 4th-polygamma function and :math:`\zeta` the
Riemann zeta function.
Parameters
----------
N : complex
Mellin moment
Returns
-------
S_5 : complex
Harmonic sum :math:`S_5(N)`
See Also
--------
ekore.harmonics.polygamma.cern_polygamma : :math:`\psi_k(N)`
"""
return zeta5 + 1.0 / 24.0 * cern_polygamma(N + 1.0, 4)
[docs]
@nb.njit(cache=True)
def Sm5(N, hS5, hS5mh, hS5h, is_singlet=None):
r"""Analytic continuation of harmonic sum :math:`S_{-5}(N)`.
.. math::
S_{-5}(N) = \sum\limits_{j=1}^N \frac {(-1)^j} {j^5}
Parameters
----------
N : complex
Mellin moment
hS5: complex
Harmonic sum :math:`S_{5}(N)`
hS5mh: complex
Harmonic sum :math:`S_{5}((N-1)/2)`
hS5h: complex
Harmonic sum :math:`S_{5}(N/2)`
is_singlet: bool, None
symmetry factor: True for singlet like quantities (:math:`\eta=(-1)^N =
1`), False for non-singlet like quantities (:math:`\eta=(-1)^N=-1`)
Returns
-------
Sm5 : complex
Harmonic sum :math:`S_{-5}(N)`
See Also
--------
eko.harmonic.w5.S5 : :math:`S_5(N)`
"""
if is_singlet is None:
return (
1 / 2**4 * ((1 - (-1) ** N) / 2 * hS5mh + ((-1) ** N + 1) / 2 * hS5h) - hS5
)
if is_singlet:
return 1 / 2**4 * hS5h - hS5
return 1 / 2**4 * hS5mh - hS5