# Fungrim entry: 8a316c

$\theta_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}$
Assumptions:$\tau \in \mathbb{H}$
TeX:
\theta_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \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("8a316c"),
Formula(Equal(Pow(JacobiTheta(3, 0, tau), 4), Where(Add(Add(1, Mul(8, Sum(Div(Mul(Mul(2, n), Pow(q, Mul(2, n))), Add(1, Pow(q, Mul(2, n)))), For(n, 0, Infinity)))), Mul(8, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Sub(1, Pow(q, Add(Mul(2, n), 1)))), For(n, 0, 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.

2020-01-31 18:09:28.494564 UTC