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Fungrim entry: d6a799

θ2 ⁣(z,τ)=2eπiτ/4cos ⁣(πz)n=1(1q2n)(1+2q2ncos ⁣(2πz)+q4n)   where q=eπiτ\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}

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
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Coscos(z)\cos(z) Cosine
Productnf(n)\prod_{n} f(n) Product
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(2, z, tau), Where(Mul(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Cos(Mul(Pi, z))), Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Add(1, Mul(Mul(2, Pow(q, Mul(2, n))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Mul(4, n)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    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