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Fungrim entry: 13d2a1

θ3 ⁣(z,τ)=n=qn2w2n=n=1(1q2n)(1+q2n1w2)(1+q2n1w2)   where q=eπiτ,w=eπiz\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {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}
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Infinity\infty Positive infinity
Productnf(n)\prod_{n} f(n) Product
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("13d2a1"),
    Formula(Where(Equal(JacobiTheta(3, z, tau), Sum(Mul(Pow(q, Pow(n, 2)), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity)), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, 2)))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), 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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2019-11-19 15:10:20.037976 UTC