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

Specific values of Jacobi theta functions