Fungrim entry: 5f9e54

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

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) + \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \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
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("5f9e54"),
    Formula(Equal(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, w, tau)), Add(Mul(JacobiTheta(1, Add(z, w), Mul(2, tau)), JacobiTheta(4, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(4, Add(z, w), Mul(2, tau)), JacobiTheta(1, Sub(z, w), Mul(2, tau)))))),
    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