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Fungrim entry: 39b699

θ2 ⁣(z,τ)=ieπiτ/4(ww1)n=1(1q2n)(1q2nw2)(1q2nw2)   where q=eπiτ,  w=eπiz\theta_{2}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \left(w - {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n} {w}^{2}\right) \left(1 - {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\theta_{2}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \left(w - {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n} {w}^{2}\right) \left(1 - {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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

2021-03-15 19:12:00.328586 UTC