# Fungrim entry: caf10a

$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{3}\!\left(z , \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}$
Assumptions:$\tau \in \mathbb{H}$
TeX:
\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{3}\!\left(z , \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
Zeros$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$ Zeros (roots) of function
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("caf10a"),
Formula(Equal(Zeros(JacobiTheta(3, z, tau), ForElement(z, CC)), Set(Add(Parentheses(Add(m, Div(1, 2))), Mul(Add(n, Div(1, 2)), tau)), For(Tuple(m, n)), And(Element(m, ZZ), Element(n, ZZ))))),
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.

2021-03-15 19:12:00.328586 UTC