Fungrim entry: 675f23

\theta_{3}\!\left(0 , 1 + 12 i\right) = \left[\frac{{2}^{-19 / 48} \cdot {3}^{-3 / 8} {\left(2 - 3 \sqrt{2} + {3}^{5 / 4} + {3}^{3 / 4}\right)}^{1 / 3}}{{\left(\sqrt{2} - 1\right)}^{1 / 12} {\left(\sqrt{3} + 1\right)}^{1 / 6} {\left(-1 - \sqrt{3} + \sqrt{2} \left({3}^{3 / 4}\right)\right)}^{1 / 3}}\right] \theta_{3}\!\left(0 , i\right)

References:

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

TeX:

\theta_{3}\!\left(0 , 1 + 12 i\right) = \left[\frac{{2}^{-19 / 48} \cdot  {3}^{-3 / 8} {\left(2 - 3 \sqrt{2} + {3}^{5 / 4} + {3}^{3 / 4}\right)}^{1 / 3}}{{\left(\sqrt{2} - 1\right)}^{1 / 12} {\left(\sqrt{3} + 1\right)}^{1 / 6} {\left(-1 - \sqrt{3} + \sqrt{2} \left({3}^{3 / 4}\right)\right)}^{1 / 3}}\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("675f23"),
    Formula(Equal(JacobiTheta(3, 0, Add(1, Mul(12, ConstI))), Mul(Brackets(Div(Mul(Mul(Pow(2, Neg(Div(19, 48))), Pow(3, Neg(Div(3, 8)))), Pow(Add(Add(Sub(2, Mul(3, Sqrt(2))), Pow(3, Div(5, 4))), Pow(3, Div(3, 4))), Div(1, 3))), Mul(Mul(Pow(Sub(Sqrt(2), 1), Div(1, 12)), Pow(Add(Sqrt(3), 1), Div(1, 6))), Pow(Add(Sub(-1, Sqrt(3)), Mul(Sqrt(2), Parentheses(Pow(3, Div(3, 4))))), Div(1, 3))))), 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