"""Weight 2 harmonic sums."""
import numba as nb
from eko.constants import zeta2
from .polygamma import cern_polygamma
[docs]
@nb.njit(cache=True)
def S2(N):
r"""Compute the harmonic sum :math:`S_2(N)`.
.. math::
S_2(N) = \sum\limits_{j=1}^N \frac 1 {j^2} = -\psi_1(N+1)+\zeta(2)
with :math:`\psi_1(N)` the trigamma function and :math:`\zeta` the
Riemann zeta function.
Parameters
----------
N : complex
Mellin moment
Returns
-------
S_2 : complex
Harmonic sum :math:`S_2(N)`
See Also
--------
ekore.harmonics.polygamma.cern_polygamma : :math:`\psi_k(N)`
"""
return -cern_polygamma(N + 1.0, 1) + zeta2
[docs]
@nb.njit(cache=True)
def Sm2(N, hS2, hS2mh, hS2h, is_singlet=None):
r"""Analytic continuation of harmonic sum :math:`S_{-2}(N)`.
.. math::
S_{-2}(N) = \sum\limits_{j=1}^N \frac {(-1)^j}{j^2}
Parameters
----------
N : complex
Mellin moment
hS2: complex
Harmonic sum :math:`S_{2}(N)`
hS2mh: complex
Harmonic sum :math:`S_{2}((N-1)/2)`
hS2h: complex
Harmonic sum :math:`S_{2}(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
-------
Sm2 : complex
Harmonic sum :math:`S_{-2}(N)`
See Also
--------
eko.anomalous_dimension.w2.S2 : :math:`S_2(N)`
"""
if is_singlet is None:
return 1 / 2 * ((1 - (-1) ** N) / 2 * hS2mh + ((-1) ** N + 1) / 2 * hS2h) - hS2
if is_singlet:
return 1 / 2 * hS2h - hS2
return 1 / 2 * hS2mh - hS2