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

Jacobi theta functions