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Fungrim entry: 8a316c

θ34 ⁣(0,τ)=1+8n=02nq2n1+q2n+8n=0(2n+1)q2n+11q2n+1   where q=eπiτ\theta_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \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_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \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("8a316c"),
    Formula(Equal(Pow(JacobiTheta(3, 0, tau), 4), Where(Add(Add(1, Mul(8, Sum(Div(Mul(Mul(2, n), Pow(q, Mul(2, n))), Add(1, Pow(q, Mul(2, n)))), For(n, 0, Infinity)))), Mul(8, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Sub(1, Pow(q, Add(Mul(2, n), 1)))), For(n, 0, Infinity)))), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

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2019-09-22 15:43:45.410764 UTC