Fungrim entry: 45a130

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

Assumptions:

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

TeX:

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \tau \;\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("45a130"),
    Formula(Equal(Mul(Mul(JacobiTheta(4, Add(z, w), tau), JacobiTheta(4, Sub(z, w), tau)), Pow(JacobiTheta(4, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

Topics using this entry

Argument transformations for Jacobi theta functions