# Fungrim entry: 28b4c3

$\theta_{3}\!\left(z , \tau + n\right) = \begin{cases} \theta_{3}\!\left(z , \tau\right), & n \text{ even}\\\theta_{4}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}$
TeX:
\theta_{3}\!\left(z , \tau + n\right) = \begin{cases} \theta_{3}\!\left(z , \tau\right), & n \text{ even}\\\theta_{4}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("28b4c3"),
Formula(Equal(JacobiTheta(3, z, Add(tau, n)), Cases(Tuple(JacobiTheta(3, z, tau), Even(n)), Tuple(JacobiTheta(4, z, tau), Odd(n))))),
Variables(z, tau, n),
Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

## 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