Fungrim entry: 5384f3

\theta_{3}\!\left(0 , 1 + 6 i\right) = \left[\frac{{\left(1 + \sqrt{3} + \sqrt{2} {\left(27\right)}^{1 / 4}\right)}^{1 / 3}}{{2}^{11 / 24} \cdot {3}^{3 / 8} {\left(\sqrt{3} - 1\right)}^{1 / 6}}\right] \theta_{3}\!\left(0 , i\right)

References:

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

TeX:

\theta_{3}\!\left(0 , 1 + 6 i\right) = \left[\frac{{\left(1 + \sqrt{3} + \sqrt{2} {\left(27\right)}^{1 / 4}\right)}^{1 / 3}}{{2}^{11 / 24} \cdot  {3}^{3 / 8} {\left(\sqrt{3} - 1\right)}^{1 / 6}}\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("5384f3"),
    Formula(Equal(JacobiTheta(3, 0, Add(1, Mul(6, ConstI))), Mul(Brackets(Div(Pow(Add(Add(1, Sqrt(3)), Mul(Sqrt(2), Pow(Parentheses(27), Div(1, 4)))), Div(1, 3)), Mul(Mul(Pow(2, Div(11, 24)), Pow(3, Div(3, 8))), Pow(Sub(Sqrt(3), 1), Div(1, 6))))), 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