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Fungrim entry: a0ba58

θ3 ⁣(0,τ)θ4 ⁣(0,τ)=n=1(1+q2n11q2n1)2   where q=eπiτ\frac{\theta_{3}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)} = \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n - 1}}{1 - {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:τH\tau \in \mathbb{H}
\frac{\theta_{3}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)} = \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n - 1}}{1 - {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

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

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2021-03-15 19:12:00.328586 UTC