Fungrim entry: 4d26ec

\theta_{4}^{8}\!\left(0, \tau\right) = 1 + 16 \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n} {n}^{3} {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{4}^{8}\!\left(0, \tau\right) = 1 + 16 \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n} {n}^{3} {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`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("4d26ec"),
    Formula(Equal(Pow(JacobiTheta(4, 0, tau), 8), Where(Add(1, Mul(16, Sum(Div(Mul(Mul(Pow(-1, n), Pow(n, 3)), Pow(q, n)), Sub(1, Pow(q, n))), 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