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Fungrim entry: 4d26ec

θ48 ⁣(0,τ)=1+16n=1(1)nn3qn1qn   where q=eπiτ\theta_{4}^{8}\!\left(0, \tau\right) = 1 + 16 \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n} {n}^{3} {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:τH\tau \in \mathbb{H}
TeX:
\theta_{4}^{8}\!\left(0, \tau\right) = 1 + 16 \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n} {n}^{3} {q}^{n}}{1 - {q}^{n}}\; \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("4d26ec"),
    Formula(Equal(Pow(JacobiTheta(4, 0, tau), 8), Where(Add(1, Mul(16, Sum(Div(Mul(Mul(Pow(-1, n), Pow(n, 3)), Pow(q, n)), Sub(1, Pow(q, n))), For(n, 1, 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.

2019-09-22 15:43:45.410764 UTC