Fungrim entry: ed4ce5

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	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("ed4ce5"),
    Formula(Equal(JacobiTheta(1, z, tau), Sum(Exp(Mul(Mul(Pi, ConstI), Sub(Add(Add(Mul(Pow(Add(n, Div(1, 2)), 2), tau), Mul(Add(Mul(2, n), 1), z)), n), Div(1, 2)))), For(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Series and product representations of Jacobi theta functions