# Fungrim entry: d2f183

$\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \tau\right)}$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}$
TeX:
\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \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
Pi$\pi$ The constant pi (3.14...)
Sin$\sin(z)$ Sine
Product$\prod_{n} f(n)$ Product
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("d2f183"),
Formula(Equal(JacobiTheta(1, z, tau), Mul(Mul(Div(JacobiTheta(1, 0, tau, 1), Pi), Sin(Mul(Pi, z))), Product(Div(Mul(Sin(Mul(Pi, Add(Mul(n, tau), z))), Sin(Mul(Pi, Sub(Mul(n, tau), z)))), Pow(Sin(Mul(Mul(Pi, n), 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.

2019-12-30 15:00:46.909060 UTC