Fungrim entry: 669765

\theta_{3}\!\left(0 , 6 i\right) = \left[\frac{{\left(-4 + 3 \sqrt{2} + {3}^{5 / 4} + 2 \sqrt{3} - {3}^{3 / 4} + 2 \sqrt{2} \left({3}^{3 / 4}\right)\right)}^{1 / 3}}{2 \left({3}^{3 / 8}\right) {\left(\left(\sqrt{2} - 1\right) \left(\sqrt{3} - 1\right)\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 , 6 i\right) = \left[\frac{{\left(-4 + 3 \sqrt{2} + {3}^{5 / 4} + 2 \sqrt{3} - {3}^{3 / 4} + 2 \sqrt{2} \left({3}^{3 / 4}\right)\right)}^{1 / 3}}{2 \left({3}^{3 / 8}\right) {\left(\left(\sqrt{2} - 1\right) \left(\sqrt{3} - 1\right)\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("669765"),
    Formula(Equal(JacobiTheta(3, 0, Mul(6, ConstI)), Mul(Brackets(Div(Pow(Add(Sub(Add(Add(Add(-4, Mul(3, Sqrt(2))), Pow(3, Div(5, 4))), Mul(2, Sqrt(3))), Pow(3, Div(3, 4))), Mul(Mul(2, Sqrt(2)), Parentheses(Pow(3, Div(3, 4))))), Div(1, 3)), Mul(Mul(2, Parentheses(Pow(3, Div(3, 8)))), Pow(Mul(Sub(Sqrt(2), 1), 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