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Fungrim entry: f1f42f

θ2 ⁣(0,τ)θ3 ⁣(0,τ)=2eπiτ/4n=1(1+q2n1+q2n1)2   where q=eπiτ\frac{\theta_{2}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} = 2 {e}^{\pi i \tau / 4} \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n}}{1 + {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:τH\tau \in \mathbb{H}
TeX:
\frac{\theta_{2}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} = 2 {e}^{\pi i \tau / 4} \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n}}{1 + {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Productnf ⁣(n)\prod_{n} f\!\left(n\right) Product
Powab{a}^{b} Power
Infinity\infty Positive infinity
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("f1f42f"),
    Formula(Equal(Div(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), Where(Mul(Mul(2, Exp(Div(Mul(Mul(ConstPi, ConstI), tau), 4))), Product(Pow(Div(Add(1, Pow(q, Mul(2, n))), Add(1, Pow(q, Sub(Mul(2, n), 1)))), 2), Tuple(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

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2019-09-20 18:07:53.062439 UTC