Fungrim entry: 13d2a1

\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \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_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Product`	$\prod_{n} f(n)$	Product
`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("13d2a1"),
    Formula(Where(Equal(JacobiTheta(3, z, tau), Sum(Mul(Pow(q, Pow(n, 2)), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity)), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, 2)))), Add(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))))

Fungrim entry: 13d2a1

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