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
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("caf10a"),
    Formula(Equal(Zeros(JacobiTheta3(z, tau), z, Element(z, CC)), SetBuilder(Add(Parentheses(Add(m, Div(1, 2))), Mul(Add(n, Div(1, 2)), tau)), Tuple(m, n), And(Element(m, ZZ), Element(n, ZZ))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

Topics using this entry

Jacobi theta functions