# Fungrim entry: 3fb309

$\theta_{3}\!\left(\frac{n}{4} , i\right) = \begin{cases} \theta_{3}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\\theta_{4}\!\left(0 , i\right), & n \equiv 2 \pmod {4}\\\left[{2}^{-7 / 16} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}$
Assumptions:$n \in \mathbb{Z}$
TeX:
\theta_{3}\!\left(\frac{n}{4} , i\right) = \begin{cases} \theta_{3}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\\theta_{4}\!\left(0 , i\right), & n \equiv 2 \pmod {4}\\\left[{2}^{-7 / 16} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
ConstI$i$ Imaginary unit
Pow${a}^{b}$ Power
Sqrt$\sqrt{z}$ Principal square root
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("3fb309"),
Formula(Equal(JacobiTheta(3, Div(n, 4), ConstI), Cases(Tuple(JacobiTheta(3, 0, ConstI), CongruentMod(n, 0, 4)), Tuple(JacobiTheta(4, 0, ConstI), CongruentMod(n, 2, 4)), Tuple(Mul(Brackets(Mul(Pow(2, Neg(Div(7, 16))), Pow(Add(Sqrt(2), 1), Div(1, 4)))), JacobiTheta(3, 0, ConstI)), Otherwise)))),
Variables(n),
Assumptions(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