# Fungrim entry: 13d2a1

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

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
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
Product$\prod_{n} f(n)$ Product
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("13d2a1"),
Formula(Where(Equal(JacobiTheta(3, z, tau), Sum(Mul(Pow(q, Pow(n, 2)), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity)), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, 2)))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, -2)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
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.

2020-04-08 16:14:44.404316 UTC