Fungrim entry: f96eac

Symbol: JacobiTheta —

\theta_{j}\!\left(z , \tau\right)

— Jacobi theta function

JacobiTheta(j, z, tau), rendered as

\theta_{j}\!\left(z , \tau\right)

, denotes a Jacobi theta function. There are four Jacobi theta functions, identified by the index

j \in \left\{1, 2, 3, 4\right\}

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.

The values of the Jacobi theta functions at

z = 0

are known as theta constants.

Called with four arguments, JacobiTheta(j, z, tau, r), rendered as

\theta'_{j}\!\left(z , \tau\right)

\theta''_{j}\!\left(z , \tau\right)

\theta'''_{j}\!\left(z , \tau\right)

(

1 \le r \le 3

), or

\theta^{(r)}_{j}\!\left(z , \tau\right)

, 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 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

\pi

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

q = {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
$j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$	$\theta_{j}\!\left(z , \tau\right) \in \mathbb{C}$
$j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$	$\theta^{(r)}_{j}\!\left(z , \tau\right) \in \mathbb{C}$

Table data:

\left(P, Q\right)

such that

\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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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", Pi, "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(Pi, 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

Jacobi theta functions