ekore.harmonics package
Module containing the harmonics sums implementation.
Definitions are coming from [Blu00, Blu09, Mus17].
Submodules
ekore.harmonics.cache module
Caching harmonic sums across ekore.
- ekore.harmonics.cache.update(func, key, cache, n)[source]
Compute simple harmonics if not yet in cache.
- ekore.harmonics.cache.get(key: int, cache: _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], n: complex, is_singlet: bool | None = None) complex[source]
Retrieve an element of the cache.
- Parameters:
key – harmonic sum key
cache – cache list holding all elements
n – complex evaluation point
is_singlet – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
requested harmonic sum evaluated at n
- Return type:
ekore.harmonics.g_functions module
Auxilary functions for harmonics sums of weight = 3,4.
Implementations of some Mellin transformations \(g_k(N)\) [Mus17] appearing in the analytic continuation of harmonics sums of weight = 3,4.
- ekore.harmonics.g_functions.mellin_g3(N, S1)[source]
Compute the Mellin transform of \(\text{Li}_2(x)/(1+x)\).
This function appears in the analytic continuation of the harmonic sum \(S_{-2,1}(N)\) which in turn appears in the NLO anomalous dimension (see Harmonic Sums).
- ekore.harmonics.g_functions.mellin_g4(N)[source]
Compute the Mellin transform of \(\text{Li}_2(-x)/(1+x)\).
Implementation and definition in B.5.25 of [Mus17] or in 61 of [Blu00], but none of them is fully correct.
- ekore.harmonics.g_functions.mellin_g5(N, S1, S2)[source]
Compute the Mellin transform of \((\text{Li}_2(x)ln(x))/(1+x)\).
Implementation and definition in B.5.26 of [Mus17] or in 62 of [Blu00], but none of them is fully correct.
- ekore.harmonics.g_functions.mellin_g6(N, S1)[source]
Compute the Mellin transform of \(\text{Li}_3(x)/(1+x)\).
Implementation and definition in B.5.27 of [Mus17] or in 63 of [Blu00], but none of them is fully correct.
- ekore.harmonics.g_functions.mellin_g8(N, S1, S2)[source]
Compute the Mellin transform of \(S_{1,2}(x)/(1+x)\).
Implementation and definition in B.5.29 of [Mus17] or in 65 of [Blu00], but none of them is fully correct.
- ekore.harmonics.g_functions.mellin_g18(N, S1, S2)[source]
Compute the Mellin transform of \(-(\text{Li}_2(x) - \zeta_2)/(1-x)\).
Implementation and definition in 124 of [Blu00]
Note: comparing to [Blu00], we believe [Mus17] was not changing the notations of \(P^{(1)}_{2}\) to \(P^{(1)}_{1}\). So we implement eq 124 of [Blu00] but using [Mus17] notation.
- ekore.harmonics.g_functions.mellin_g19(N, S1)[source]
Compute the Mellin transform of \(-(\text{Li}_2(-x) + \zeta_2/2)/(1-x)\).
Implementation and definition in B.5.40 of [Mus17] or in 125 of [Blu00], but none of them is fully correct.
- ekore.harmonics.g_functions.mellin_g21(N, S1, S2, S3)[source]
Compute the Mellin transform of \(-(S_{1,2}(x) - \zeta_3)/(1-x)\).
Implementation and definition in B.5.42 of [Mus17].
Note: comparing to [Blu00], we believe [Mus17] was not changing the notations of \(P^{(3)}_{2}\) to \(P^{(3)}_{1}\) and \(P^{(3)}_{3}\) to \(P^{(3)}_{2}\). So we implement 127 of [Blu00] but using [Mus17] notation.
- ekore.harmonics.g_functions.mellin_g22(N, S1, S2, S3)[source]
Compute the Mellin transform of \(-(\text{Li}_2(x) ln(x))/(1-x)\).
Implementation and definition in B.5.43 of [Mus17].
Note: comparing to [Blu00], we believe [Mus17] was not changing the notations of \(P^{(1)}_{2}\) to \(P^{(1)}_{1}\) So we implement 128 of [Blu00] but using [Mus17] notation.
ekore.harmonics.log_functions module
Implementation of Mellin transformation of logarithms.
We provide transforms of:
\((1-x)\ln^k(1-x), \quad k = 1,2,3\)
\(\ln^k(1-x), \quad k = 1,3,4,5\)
- ekore.harmonics.log_functions.lm13m1(n, S1, S2, S3)[source]
Mellin transform of \((1-x)\ln^3(1-x)\).
- ekore.harmonics.log_functions.lm14m1(n, S1, S2, S3, S4)[source]
Mellin transform of \((1-x)\ln^4(1-x)\).
- ekore.harmonics.log_functions.lm15m1(n, S1, S2, S3, S4, S5)[source]
Mellin transform of \((1-x)\ln^5(1-x)\).
- Parameters:
- Returns:
\(\mathcal{M}[(1-x)\ln^5(1-x)](N)\)
- Return type:
- ekore.harmonics.log_functions.lm15(n, S1, S2, S3, S4, S5)[source]
Mellin transform of \(\ln^5(1-x)\).
- Parameters:
- Returns:
\(\mathcal{M}[\ln^5(1-x)](N)\)
- Return type:
- ekore.harmonics.log_functions.lm13m2(n, S1, S2, S3)[source]
Mellin transform of \((1-x)^2\ln^3(1-x)\).
ekore.harmonics.polygamma module
Polygamma and harmonic sums implementation.
The functions are described in Mellin space.
- ekore.harmonics.polygamma.cern_polygamma(Z, K)[source]
Compute the polygamma functions \(\psi_k(z)\).
Reimplementation of
WPSIPG(C317) in CERNlib [Kol72].Note that the SciPy implementation
scipy.special.digammadoes not allow for complex inputs.
- ekore.harmonics.polygamma.recursive_harmonic_sum(base_value, n, iterations, weight)[source]
Recursive computation of harmonic sums.
Compute the harmonic sum \(S_{w}(N+k)\) stating from the value \(S_{w}(N)\) via the recurrence relations.
ekore.harmonics.w1 module
Weight 1 harmonic sums.
- ekore.harmonics.w1.S1(N)[source]
Compute the harmonic sum \(S_1(N)\).
\[S_1(N) = \sum\limits_{j=1}^N \frac 1 j = \psi_0(N+1)+\gamma_E\]with \(\psi_0(N)\) the digamma function and \(\gamma_E\) the Euler-Mascheroni constant.
- Parameters:
N (complex) – Mellin moment
- Returns:
S_1 – (simple) Harmonic sum \(S_1(N)\)
- Return type:
See also
ekore.harmonics.polygamma.cern_polygamma\(\psi_k(N)\)
- ekore.harmonics.w1.Sm1(N, hS1, hS1mh, hS1h, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-1}(N)\).
\[S_{-1}(N) = \sum\limits_{j=1}^N \frac {(-1)^j} j\]- Parameters:
N (complex) – Mellin moment
hS1 (complex) – Harmonic sum \(S_{1}(N)\)
hS1mh (complex) – Harmonic sum \(S_{1}((N-1)/2)\)
hS1h (complex) – Harmonic sum \(S_{1}(N/2)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm1 – Harmonic sum \(S_{-1}(N)\)
- Return type:
See also
eko.anomalous_dimension.w1.S1\(S_1(N)\)
ekore.harmonics.w2 module
Weight 2 harmonic sums.
- ekore.harmonics.w2.S2(N)[source]
Compute the harmonic sum \(S_2(N)\).
\[S_2(N) = \sum\limits_{j=1}^N \frac 1 {j^2} = -\psi_1(N+1)+\zeta(2)\]with \(\psi_1(N)\) the trigamma function and \(\zeta\) the Riemann zeta function.
See also
ekore.harmonics.polygamma.cern_polygamma\(\psi_k(N)\)
- ekore.harmonics.w2.Sm2(N, hS2, hS2mh, hS2h, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-2}(N)\).
\[S_{-2}(N) = \sum\limits_{j=1}^N \frac {(-1)^j}{j^2}\]- Parameters:
N (complex) – Mellin moment
hS2 (complex) – Harmonic sum \(S_{2}(N)\)
hS2mh (complex) – Harmonic sum \(S_{2}((N-1)/2)\)
hS2h (complex) – Harmonic sum \(S_{2}(N/2)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm2 – Harmonic sum \(S_{-2}(N)\)
- Return type:
See also
eko.anomalous_dimension.w2.S2\(S_2(N)\)
ekore.harmonics.w3 module
Weight 3 harmonic sums.
- ekore.harmonics.w3.S3(N)[source]
Compute the harmonic sum \(S_3(N)\).
\[S_3(N) = \sum\limits_{j=1}^N \frac 1 {j^3} = \frac 1 2 \psi_2(N+1)+\zeta(3)\]with \(\psi_2(N)\) the 2nd-polygamma function and \(\zeta\) the Riemann zeta function.
See also
ekore.harmonics.polygamma.cern_polygamma\(\psi_k(N)\)
- ekore.harmonics.w3.Sm3(N, hS3, hS3mh, hS3h, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-3}(N)\).
\[S_{-3}(N) = \sum\limits_{j=1}^N \frac {(-1)^j} {j^3}\]- Parameters:
N (complex) – Mellin moment
hS3 (complex) – Harmonic sum \(S_{3}(N)\)
hS3mh (complex) – Harmonic sum \(S_{3}((N-1)/2)\)
hS3h (complex) – Harmonic sum \(S_{3}(N/2)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm3 – Harmonic sum \(S_{-3}(N)\)
- Return type:
See also
ekore.harmonics.w3.S3\(S_3(N)\)
- ekore.harmonics.w3.Sm21(N, S1, Sm1, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-2,1}(N)\).
As implemented in B.5.75 of [Mus17] and 22 of [Blu00].
- Parameters:
- Returns:
Sm21 – Harmonic sum \(S_{-2,1}(N)\)
- Return type:
- ekore.harmonics.w3.S2m1(N, S2, Sm1, Sm2, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{2,-1}(N)\).
As implemented in B.5.76 of [Mus17] and 23 of [Blu00].
- Parameters:
N (complex) – Mellin moment
S2 (complex) – Harmonic sum \(S_{2}(N)\)
Sm1 (complex) – Harmonic sum \(S_{-1}(N)\)
Sm2 (complex) – Harmonic sum \(S_{-2}(N)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
S2m1 – Harmonic sum \(S_{2,-1}(N)\)
- Return type:
ekore.harmonics.w4 module
Weight 4 harmonic sums.
- ekore.harmonics.w4.S4(N)[source]
Compute the harmonic sum \(S_4(N)\).
\[S_4(N) = \sum\limits_{j=1}^N \frac 1 {j^4} = - \frac 1 6 \psi_3(N+1)+\zeta(4)\]with \(\psi_3(N)\) the 3rd-polygamma function and \(\zeta\) the Riemann zeta function.
See also
ekore.harmonics.polygamma.cern_polygamma\(\psi_k(N)\)
- ekore.harmonics.w4.Sm4(N, hS4, hS4mh, hS4h, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-4}(N)\).
\[S_{-4}(N) = \sum\limits_{j=1}^N \frac {(-1)^j} {j^4}\]- Parameters:
N (complex) – Mellin moment
hS4 (complex) – Harmonic sum \(S_{4}(N)\)
hS4mh (complex) – Harmonic sum \(S_{4}((N-1)/2)\)
hS4h (complex) – Harmonic sum \(S_{4}(N/2)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm4 – Harmonic sum \(S_{-4}(N)\)
- Return type:
See also
eko.anomalous_dimension.w4.S4\(S_4(N)\)
- ekore.harmonics.w4.Sm31(N, S1, Sm1, Sm2, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-3,1}(N)\).
As implemented in B.5.93 of [Mus17] and 25 of cite:Bl_mlein_2000.
- Parameters:
N (complex) – Mellin moment
S1 (complex) – Harmonic sum \(S_{1}(N)\)
Sm1 (complex) – Harmonic sum \(S_{-1}(N)\)
Sm2 (complex) – Harmonic sum \(S_{-2}(N)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm31 – Harmonic sum \(S_{-3,1}(N)\)
- Return type:
See also
ekore.harmonics.g_functions.mellin_g6\(g_6(N)\)
- ekore.harmonics.w4.Sm22(N, S1, S2, Sm2, Sm31, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-2,2}(N)\).
As implemented in B.5.94 of [Mus17] and 24 of cite:Bl_mlein_2000.
- Parameters:
N (complex) – Mellin moment
S1 (complex) – Harmonic sum \(S_{1}(N)\)
S2 (complex) – Harmonic sum \(S_{2}(N)\)
Sm2 (complex) – Harmonic sum \(S_{-2}(N)\)
Sm31 (complex) – Harmonic sum \(S_{-3,1}(N)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm22 – Harmonic sum \(S_{-2,2}(N)\)
- Return type:
See also
ekore.harmonics.g_functions.mellin_g5\(g_5(N)\)
- ekore.harmonics.w4.Sm211(N, S1, S2, Sm1, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-2,1,1}(N)\).
As implemented in B.5.104 of [Mus17] and 27 of cite:Bl_mlein_2000.
- Parameters:
N (complex) – Mellin moment
S1 (complex) – Harmonic sum \(S_{1}(N)\)
S2 (complex) – Harmonic sum \(S_{2}(N)\)
Sm1 (complex) – Harmonic sum \(S_{-1}(N)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm221 – Harmonic sum \(S_{-2,1,1}(N)\)
- Return type:
See also
ekore.harmonics.g_functions.mellin_g8\(g_8(N)\)
- ekore.harmonics.w4.S211(N, S1, S2, S3)[source]
Analytic continuation of harmonic sum \(S_{2,1,1}(N)\).
As implemented in B.5.115 of [Mus17] and 40 of cite:Bl_mlein_2000.
- Parameters:
- Returns:
S211 – Harmonic sum \(S_{2,1,1}(N)\)
- Return type:
See also
ekore.harmonics.g_functions.mellin_g21\(g_21(N)\)
- ekore.harmonics.w4.S31(N, S1, S2, S3, S4)[source]
Analytic continuation of harmonic sum \(S_{3,1}(N)\).
As implemented in B.5.99 of [Mus17] and 41 of cite:Bl_mlein_2000.
- Parameters:
- Returns:
S31 – Harmonic sum \(S_{3,1}(N)\)
- Return type:
See also
ekore.harmonics.g_functions.mellin_g22\(g_22(N)\)
ekore.harmonics.w5 module
Weight 5 harmonic sums.
- ekore.harmonics.w5.S5(N)[source]
Compute the harmonic sum \(S_5(N)\).
\[S_5(N) = \sum\limits_{j=1}^N \frac 1 {j^5} = \frac 1 24 \psi_4(N+1)+\zeta(5)\]with \(\psi_4(N)\) the 4th-polygamma function and \(\zeta\) the Riemann zeta function.
See also
ekore.harmonics.polygamma.cern_polygamma\(\psi_k(N)\)
- ekore.harmonics.w5.Sm5(N, hS5, hS5mh, hS5h, is_singlet=None)[source]
Analytic continuation of harmonic sum \(S_{-5}(N)\).
\[S_{-5}(N) = \sum\limits_{j=1}^N \frac {(-1)^j} {j^5}\]- Parameters:
N (complex) – Mellin moment
hS5 (complex) – Harmonic sum \(S_{5}(N)\)
hS5mh (complex) – Harmonic sum \(S_{5}((N-1)/2)\)
hS5h (complex) – Harmonic sum \(S_{5}(N/2)\)
is_singlet (bool, None) – symmetry factor: True for singlet like quantities (\(\eta=(-1)^N = 1\)), False for non-singlet like quantities (\(\eta=(-1)^N=-1\))
- Returns:
Sm5 – Harmonic sum \(S_{-5}(N)\)
- Return type:
See also
eko.harmonic.w5.S5\(S_5(N)\)