Fungrim entry: 580ba0

\theta_3\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_3\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`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("580ba0"),
    Formula(Equal(JacobiTheta3(z, tau), Sum(Exp(Mul(Mul(ConstPi, ConstI), Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z)))), Tuple(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Jacobi theta functions