Fungrim entry: 7f9273

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

References:

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

TeX:

\theta_{3}\!\left(0 , \frac{i}{4}\right) = \left[\frac{1 + {2}^{-1 / 4}}{\sqrt{1 + \sqrt{2}}} \sqrt{\frac{\sqrt{2} + 1}{2}} \cdot  {2}^{1 / 2}\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("7f9273"),
    Formula(Equal(JacobiTheta(3, 0, Div(ConstI, 4)), Mul(Brackets(Mul(Mul(Div(Add(1, Pow(2, Neg(Div(1, 4)))), Sqrt(Add(1, Sqrt(2)))), Sqrt(Div(Add(Sqrt(2), 1), 2))), Pow(2, Div(1, 2)))), 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