# Fungrim entry: 77aed2

$\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}

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
Product$\prod_{n} f(n)$ Product
Pow${a}^{b}$ Power
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("77aed2"),
Formula(Equal(JacobiTheta(3, z, tau), Where(Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Add(1, Mul(Mul(2, Pow(q, Sub(Mul(2, n), 1))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Sub(Mul(4, n), 2)))), For(n, 1, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
Variables(z, tau),
Assumptions(And(Element(z, CC), 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.

2021-03-15 19:12:00.328586 UTC