# Fungrim entry: 580ba0

$\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\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("580ba0"),
Formula(Equal(JacobiTheta(3, z, tau), Sum(Exp(Mul(Mul(Pi, ConstI), Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z)))), For(n, Neg(Infinity), Infinity)))),
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