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

Specific values of Jacobi theta functions