Fungrim entry: 7e0002

\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right)}{2 \theta_{3}\!\left(0 , 2 \tau\right)}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right)}{2 \theta_{3}\!\left(0 , 2 \tau\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7e0002"),
    Formula(Equal(JacobiTheta(3, Mul(2, z), Mul(2, tau)), Div(Add(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2)), Mul(2, JacobiTheta(3, 0, Mul(2, tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Lattice transformations for Jacobi theta functions