# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC