# Fungrim entry: d923de

$\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {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}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("d923de"),
Formula(Equal(JacobiTheta(4, z, tau), Where(Sum(Mul(Mul(Pow(-1, n), 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),
Assumptions(And(Element(z, CC), Element(tau, HH))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC