# Fungrim entry: 1842d9

$\theta^{(r)}_{4}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$
TeX:
\theta^{(r)}_{4}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \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
Sum$\sum_{n} f(n)$ Sum
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
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("1842d9"),
Formula(Equal(JacobiTheta(4, z, tau, r), Where(Mul(Pow(Mul(Mul(2, Pi), ConstI), r), Sum(Mul(Mul(Mul(Pow(-1, n), Pow(n, r)), Pow(q, Pow(n, 2))), Pow(w, Mul(2, n))), 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC