Fungrim entry: e6d333

\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \frac{\tau}{2}\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \frac{\tau}{2}\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("e6d333"),
    Formula(Equal(JacobiTheta(3, z, Div(tau, 2)), Div(Sub(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(1, z, tau), 2)), JacobiTheta(4, 0, Div(tau, 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Lattice transformations for Jacobi theta functions