Fungrim entry: 42d832

\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("42d832"),
    Formula(Equal(JacobiTheta(2, z, tau, r), Where(Mul(Mul(Pow(Mul(Pi, ConstI), r), Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Mul(Pow(Add(Mul(2, n), 1), r), Pow(q, Mul(n, Add(n, 1)))), Pow(w, Add(Mul(2, n), 1))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau, r),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

Topics using this entry

Series and product representations of Jacobi theta functions