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Fungrim entry: 77aed2

θ3 ⁣(z,τ)=n=1(1q2n)(1+2q2n1cos ⁣(2πz)+q4n2)   where q=eπiτ\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\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
Productnf(n)\prod_{n} f(n) Product
Powab{a}^{b} Power
Coscos(z)\cos(z) Cosine
Piπ\pi The constant pi (3.14...)
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(3, z, tau), Where(Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Add(1, Mul(Mul(2, Pow(q, Sub(Mul(2, n), 1))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Sub(Mul(4, n), 2)))), 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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2021-03-15 19:12:00.328586 UTC