Fungrim entry: ed0756

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \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
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ed0756"),
    Formula(Equal(JacobiTheta(1, z, tau), Mul(Mul(Neg(ConstI), Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(3, Add(Add(z, Div(1, 2)), Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Argument transformations for Jacobi theta functions