Fungrim entry: 27b169

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

Assumptions:

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

TeX:

\theta_{1}\!\left(4 z , 4 \tau\right) = \frac{\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(\frac{1}{4} - z , \tau\right) \theta_{1}\!\left(\frac{1}{4} + z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \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
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("27b169"),
    Formula(Equal(JacobiTheta(1, Mul(4, z), Mul(4, tau)), Div(Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(1, Sub(Div(1, 4), z), tau)), JacobiTheta(1, Add(Div(1, 4), z), tau)), JacobiTheta(2, z, tau)), Mul(Mul(JacobiTheta(3, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(3, Div(1, 4), tau))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Lattice transformations for Jacobi theta functions