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Fungrim entry: 1cec67

θ44 ⁣(0,τ)θ24 ⁣(0,τ)=124n=0(2n+1)q2n+11+q2n+1   where q=eπiτ\theta_{4}^{4}\!\left(0, \tau\right) - \theta_{2}^{4}\!\left(0, \tau\right) = 1 - 24 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:τH\tau \in \mathbb{H}
TeX:
\theta_{4}^{4}\!\left(0, \tau\right) - \theta_{2}^{4}\!\left(0, \tau\right) = 1 - 24 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("1cec67"),
    Formula(Equal(Sub(Pow(JacobiTheta(4, 0, tau), 4), Pow(JacobiTheta(2, 0, tau), 4)), Where(Sub(1, Mul(24, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Add(1, Pow(q, Add(Mul(2, n), 1)))), Tuple(n, 0, Infinity)))), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-20 18:07:53.062439 UTC