Fungrim entry: 429093

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

Assumptions:

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

TeX:

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

Source code for this entry:

Entry(ID("429093"),
    Formula(Equal(JacobiTheta(4, Add(z, Mul(Div(1, 2), tau)), tau), Mul(Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), ConstI), JacobiTheta(1, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Argument transformations for Jacobi theta functions