Fungrim entry: d45548

\theta_{4}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n - 1} {w}^{2}\right) \left(1 - {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

\theta_{4}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n - 1} {w}^{2}\right) \left(1 - {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d45548"),
    Formula(Where(Equal(JacobiTheta(4, z, tau), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Sub(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, 2)))), Sub(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, -2)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Series and product representations of Jacobi theta functions