Fungrim entry: 2ba423

\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} \sin\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} \sin\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("2ba423"),
    Formula(Equal(JacobiTheta(1, z, tau), Where(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Mul(Pow(-1, n), Pow(q, Mul(n, Add(n, 1)))), Sin(Mul(Mul(Add(Mul(2, n), 1), Pi), z))), For(n, 0, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Fungrim entry: 2ba423

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