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Fungrim entry: f96eac

Symbol: JacobiTheta θj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
JacobiTheta(j, z, tau), rendered as θj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right), denotes a Jacobi theta function. There are four Jacobi theta functions, identified by the index j{1,2,3,4}j \in \left\{1, 2, 3, 4\right\}.
The input zz is called the argument and can be any complex number. The input τ\tau is called the lattice parameter and must be a complex number with positive imaginary part.
The values of the Jacobi theta functions at z=0z = 0 are known as theta constants.
Called with four arguments, JacobiTheta(j, z, tau, r), rendered as θj ⁣(z,τ)\theta'_{j}\!\left(z , \tau\right), θj ⁣(z,τ)\theta''_{j}\!\left(z , \tau\right), θj ⁣(z,τ)\theta'''_{j}\!\left(z , \tau\right) ( 1r31 \le r \le 3 ), or θj(r) ⁣(z,τ)\theta^{(r)}_{j}\!\left(z , \tau\right), represents the order rr derivative of the Jacobi theta function with respect to the argument zz. Derivatives with respect to the lattice parameter τ\tau (and mixed derivatives) can always be converted to derivatives with respect to zz, using 37e644.
The Jacobi theta functions are defined by the respective Fourier series ( 700d94, 495a98, 2f97f5, d923de ). It is important to note that Fungrim defines theta functions with a factor π\pi applied to the argument zz in the Fourier series, for uniformity with the lattice parameter τ\tau. Many authors omit this scaling factor or replace the input τ\tau by q=eπiτq = {e}^{\pi i \tau}. Other conventions exist in the mathematical literature as well, so care is required when using different reference works.
The following table lists conditions such that JacobiTheta(j, z, tau) or JacobiTheta(j, z, tau, r) is defined in Fungrim.
Domain Codomain
j{1,2,3,4}andzCandτHj \in \left\{1, 2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} θj ⁣(z,τ)C\theta_{j}\!\left(z , \tau\right) \in \mathbb{C}
j{1,2,3,4}andzCandτHandrZ0j \in \left\{1, 2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} θj(r) ⁣(z,τ)C\theta^{(r)}_{j}\!\left(z , \tau\right) \in \mathbb{C}
Table data: (P,Q)\left(P, Q\right) such that (P)    (Q)\left(P\right) \implies \left(Q\right)
References:
  • https://dlmf.nist.gov/20
  • http://functions.wolfram.com/EllipticFunctions/EllipticTheta1/introductions/JacobiThetas/
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstPiπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("f96eac"),
    SymbolDefinition(JacobiTheta, JacobiTheta(j, z, tau), "Jacobi theta function"),
    Description(SourceForm(JacobiTheta(j, z, tau)), ", rendered as", JacobiTheta(j, z, tau), ", denotes a Jacobi theta function. ", "There are four Jacobi theta functions, identified by the index", Element(j, Set(1, 2, 3, 4)), "."),
    Description("The input", z, "is called the argument and can be any complex number. ", "The input", tau, "is called the lattice parameter and must be a complex number with positive imaginary part."),
    Description("The values of the Jacobi theta functions at", Equal(z, 0), "are known as theta constants."),
    Description("Called with four arguments, ", SourceForm(JacobiTheta(j, z, tau, r)), ", rendered as", JacobiTheta(j, z, tau, 1), ", ", JacobiTheta(j, z, tau, 2), ", ", JacobiTheta(j, z, tau, 3), " (", LessEqual(1, r, 3), "), or", JacobiTheta(j, z, tau, r), ", represents the order", r, "derivative of the Jacobi theta function with respect to the argument", z, ".", "Derivatives with respect to the lattice parameter", tau, "(and mixed derivatives) can always be converted to derivatives with respect to", z, ", using", EntryReference("37e644"), "."),
    Description("The Jacobi theta functions are defined by the respective Fourier series (", EntryReference("700d94"), ", ", EntryReference("495a98"), ", ", EntryReference("2f97f5"), ", ", EntryReference("d923de"), "). ", "It is important to note that Fungrim defines theta functions with a factor", ConstPi, "applied to the argument", z, "in the Fourier series, for uniformity with the lattice parameter", tau, ". Many authors omit this scaling factor or replace the input", tau, "by", Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau))), ". Other conventions exist in the mathematical literature as well, so care is required when using different reference works."),
    Description("The following table lists conditions such that", SourceForm(JacobiTheta(j, z, tau)), "or", SourceForm(JacobiTheta(j, z, tau, r)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH)), Element(JacobiTheta(j, z, tau), CC)), Tuple(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0))), Element(JacobiTheta(j, z, tau, r), CC)))),
    References("https://dlmf.nist.gov/20", "http://functions.wolfram.com/EllipticFunctions/EllipticTheta1/introductions/JacobiThetas/"))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC