Fungrim entry: c4b16c

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

Assumptions:

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

TeX:

\theta_{3}\!\left(z , -\frac{1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{3}\!\left(\tau 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
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("c4b16c"),
    Formula(Equal(JacobiTheta(3, z, Div(-1, tau)), Mul(Mul(Sqrt(Div(tau, ConstI)), Exp(Mul(Mul(Mul(Pi, ConstI), tau), Pow(z, 2)))), JacobiTheta(3, Mul(tau, z), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Fungrim entry: c4b16c

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