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# The table document for numberdb.org/T426, numberdb-data#191 proposal 7. It was
# created as a draft through the API with X-Draft: yes and is filled by
# generate.py beside it; publishing is a person's act.
Title: Values of the Jacobi theta constant $\theta_4(0,q)$
Definition: >-
For a real nome $0<q<1$, this table stores the Jacobi theta constant
$\theta_4(0,q)$, with $q=e^{i\pi\tau}$ and
$\theta_4(z,q)=1+2\sum_{n=1}^{\infty}(-1)^nq^{n^2}\cos(2nz)$
CITE{DLMFThetaNotation} CITE{DLMFThetaDefinitions}.
Keywords:
- Jacobi theta
- theta constant
- theta four
Parameters:
q:
type: Q
title: real nome
display: $q$
constraints: $0<q<1$
Comments:
comment-nome: >-
The parameter is the real nome $q$, not the lattice parameter $\tau$. In
this rectangular case $0<q<1$, the relation $q=e^{i\pi\tau}$ determines
$\tau=\log(q)/(i\pi)$ uniquely CITE{DLMFThetaNotation}.
comment-theta1: >-
The table stores a theta constant, so $z=0$ is not a parameter. There is
no nonzero companion for $\theta_1(0,q)$, since $\theta_1(z,q)$ is odd in
$z$ CITE{DLMFThetaDefinitions}.
Formulas:
formula-series: >-
$\theta_4(0,q)=1+2\sum_{n=1}^{\infty}(-1)^nq^{n^2}$.
formula-jacobi-identity: >-
$\theta_3(0,q)^4=\theta_2(0,q)^4+\theta_4(0,q)^4$
CITE{DLMFThetaIdentities}.
Programs:
program-sage:
language: Sage
code: |
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ
CBF = ComplexBallField(numberdb.bits(100, losing=80))
q = QQ(1) / QQ(10)
tau = CBF(0, -1) * CBF(q).log() / CBF.pi()
value = CBF(0).jacobi_theta(tau)[3]
assert value.imag().contains_zero()
value.real()
Similar tables:
- table: HREF{Values_of_the_elliptic_nome}[Values of the elliptic nome $q(m)$]
relation: the nome table stores the elliptic-parameter values whose theta constants recover the modulus
- table: HREF{Complete_elliptic_integral_of_the_first_kind_K}[Complete elliptic integral of the first kind $K(m)$]
relation: the theta constants and $K(m)$ are related by Jacobi's inversion formulas CITE{DLMFThetaRelations}
Links:
DLMFThetaNotation:
title: 'DLMF 20.1: Special notation for theta functions'
url: https://dlmf.nist.gov/20.1
DLMFThetaDefinitions:
title: 'DLMF 20.2: Definitions and periodic properties of theta functions'
url: https://dlmf.nist.gov/20.2
DLMFThetaIdentities:
title: 'DLMF 20.7: Identities for theta functions'
url: https://dlmf.nist.gov/20.7
DLMFThetaRelations:
title: 'DLMF 20.9: Relations of theta functions to other functions'
url: https://dlmf.nist.gov/20.9
Tags:
- special values
- special functions
Data properties:
type: R
rigour: proven
complete: 'no'
complete-note: >-
it holds $\theta_4(0,q)$ for every decimal grid point $q=j/1000$, with
$j=1,\ldots,900$
rigour details: >-
Each value was computed as a Sage complex ball with arb at
`numberdb.bits(digits, losing=80)` bits, after converting the real nome to
$\tau=\log(q)/(i\pi)$. The imaginary part was checked to contain zero, and
the real part was returned. Before the draft was filled, the run compared
every value with the defining $q$-series with an explicit tail bound and
checked CITE{formula-jacobi-identity} across the stored range.
Display properties:
number-header: $\theta_4(0,q)$