Fungrim entry: 816057

\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \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
`Product`	$\prod_{n} f(n)$	Product
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("816057"),
    Formula(Equal(JacobiTheta(3, z, tau), Mul(JacobiTheta(3, 0, tau), Product(Div(Mul(Cos(Mul(Pi, Add(Mul(Sub(n, Div(1, 2)), tau), z))), Cos(Mul(Pi, Sub(Mul(Sub(n, Div(1, 2)), tau), z)))), Pow(Cos(Mul(Pi, Mul(Sub(n, Div(1, 2)), tau))), 2)), For(n, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Series and product representations of Jacobi theta functions