Fungrim entry: 28b4c3

\theta_{3}\!\left(z , \tau + n\right) = \begin{cases} \theta_{3}\!\left(z , \tau\right), & n \text{ even}\\\theta_{4}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau + n\right) = \begin{cases} \theta_{3}\!\left(z , \tau\right), & n \text{ even}\\\theta_{4}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}

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
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("28b4c3"),
    Formula(Equal(JacobiTheta(3, z, Add(tau, n)), Cases(Tuple(JacobiTheta(3, z, tau), Even(n)), Tuple(JacobiTheta(4, z, tau), Odd(n))))),
    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