# Fungrim entry: 816057

$\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}

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
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("816057"),
Formula(Equal(JacobiTheta(3, z, tau), Mul(JacobiTheta(3, 0, tau), Product(Div(Mul(Cos(Mul(Pi, Add(Mul(Sub(n, Div(1, 2)), tau), z))), Cos(Mul(Pi, Sub(Mul(Sub(n, Div(1, 2)), tau), z)))), Pow(Cos(Mul(Pi, Mul(Sub(n, Div(1, 2)), tau))), 2)), For(n, 1, Infinity))))),
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