# Fungrim entry: dd5f43

$\theta_{2}\!\left(\frac{n}{4} , i\right) = \begin{cases} {\left(-1\right)}^{\left\lfloor \left( n + 1 \right) / 4 \right\rfloor} \theta_{4}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\0, & n \equiv 2 \pmod {4}\\{\left(-1\right)}^{\left\lfloor \left( n + 1 \right) / 4 \right\rfloor} \left[{2}^{-7 / 16} \sqrt{\sqrt{2} - 1} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}$
Assumptions:$n \in \mathbb{Z}$
TeX:
\theta_{2}\!\left(\frac{n}{4} , i\right) = \begin{cases} {\left(-1\right)}^{\left\lfloor \left( n + 1 \right) / 4 \right\rfloor} \theta_{4}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\0, & n \equiv 2 \pmod {4}\\{\left(-1\right)}^{\left\lfloor \left( n + 1 \right) / 4 \right\rfloor} \left[{2}^{-7 / 16} \sqrt{\sqrt{2} - 1} {\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("dd5f43"),
Formula(Equal(JacobiTheta(2, Div(n, 4), ConstI), Cases(Tuple(Mul(Pow(-1, Floor(Div(Add(n, 1), 4))), JacobiTheta(4, 0, ConstI)), CongruentMod(n, 0, 4)), Tuple(0, CongruentMod(n, 2, 4)), Tuple(Mul(Mul(Pow(-1, Floor(Div(Add(n, 1), 4))), Brackets(Mul(Mul(Pow(2, Neg(Div(7, 16))), Sqrt(Sub(Sqrt(2), 1))), 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