Fungrim entry: 44e8fb

\frac{1}{\pi} \frac{\theta'_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|

TeX:

\frac{1}{\pi} \frac{\theta'_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part

Source code for this entry:

Entry(ID("44e8fb"),
    Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(3, z, tau, 1), JacobiTheta(3, z, tau))), Where(Mul(4, Sum(Mul(Mul(Pow(-1, n), Div(Pow(q, n), Sub(1, Pow(q, Mul(2, n))))), Sin(Mul(Mul(Mul(2, Pi), n), z))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Mul(Div(1, 2), Abs(Im(tau)))))))

Topics using this entry

Series and product representations of Jacobi theta functions