Fungrim entry: 324483

\theta_{3}\!\left(0 , 1 + \frac{i}{2}\right) = \left[\frac{{\left(\sqrt{2} - 1\right)}^{2 / 3} {\left(4 + 3 \sqrt{2}\right)}^{1 / 12}}{{2}^{7 / 24}}\right] \theta_{3}\!\left(0 , i\right)

References:

https://doi.org/10.1016/j.jmaa.2003.12.009

TeX:

\theta_{3}\!\left(0 , 1 + \frac{i}{2}\right) = \left[\frac{{\left(\sqrt{2} - 1\right)}^{2 / 3} {\left(4 + 3 \sqrt{2}\right)}^{1 / 12}}{{2}^{7 / 24}}\right] \theta_{3}\!\left(0 , i\right)

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

Source code for this entry:

Entry(ID("324483"),
    Formula(Equal(JacobiTheta(3, 0, Add(1, Div(ConstI, 2))), Mul(Brackets(Div(Mul(Pow(Sub(Sqrt(2), 1), Div(2, 3)), Pow(Add(4, Mul(3, Sqrt(2))), Div(1, 12))), Pow(2, Div(7, 24)))), JacobiTheta(3, 0, ConstI)))),
    References("https://doi.org/10.1016/j.jmaa.2003.12.009"))

Topics using this entry

Specific values of Jacobi theta functions