Fungrim entry: 8a34d1

\theta_4\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)

Assumptions:

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

TeX:

\theta_4\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("8a34d1"),
    Formula(Equal(JacobiTheta4(z, tau), Add(1, Mul(2, Sum(Mul(Mul(Pow(-1, n), Exp(Mul(Mul(Mul(ConstPi, ConstI), Pow(n, 2)), tau))), Cos(Mul(Mul(Mul(2, n), ConstPi), z))), Tuple(n, 1, Infinity)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Jacobi theta functions