Fungrim entry: b978f0

\theta_{1}\!\left(z , \tau + 2 n\right) = {i}^{n} \theta_{1}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau + 2 n\right) = {i}^{n} \theta_{1}\!\left(z , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("b978f0"),
    Formula(Equal(JacobiTheta(1, z, Add(tau, Mul(2, n))), Mul(Pow(ConstI, n), JacobiTheta(1, z, tau)))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

Topics using this entry

Lattice transformations for Jacobi theta functions