Fungrim entry: a0ba58

\frac{\theta_{3}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)} = \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n - 1}}{1 - {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\frac{\theta_{3}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)} = \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n - 1}}{1 - {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

\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
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a0ba58"),
    Formula(Equal(Div(JacobiTheta(3, 0, tau), JacobiTheta(4, 0, tau)), Where(Product(Pow(Div(Add(1, Pow(q, Sub(Mul(2, n), 1))), Sub(1, Pow(q, Sub(Mul(2, n), 1)))), 2), For(n, 1, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

Topics using this entry

Series and product representations of Jacobi theta functions