Fungrim entry: 43fa0e

\theta_1\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_1\!\left(z, \tau\right)

Assumptions:

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

TeX:

\theta_1\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_1\!\left(z, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`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("43fa0e"),
    Formula(Equal(JacobiTheta1(Add(z, Add(m, Mul(n, tau))), tau), Mul(Mul(Pow(-1, Add(m, n)), Exp(Neg(Mul(Mul(ConstPi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z)))))), JacobiTheta1(z, tau)))),
    Variables(z, tau, m, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(m, ZZ), Element(n, ZZ))))

Topics using this entry

Jacobi theta functions