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Fungrim entry: d923de

θ4 ⁣(z,τ)=n=(1)nqn2w2n   where q=eπiτ,w=eπiz\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} {w}^{2 n}\; \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_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} {w}^{2 n}\; \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
Expez{e}^{z} Exponential function
ConstPiπ\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("d923de"),
    Formula(Equal(JacobiTheta(4, z, tau), Where(Sum(Mul(Mul(Pow(-1, n), Pow(q, Pow(n, 2))), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity)), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau))), Equal(w, Exp(Mul(Mul(ConstPi, ConstI), z)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2019-10-05 13:11:19.856591 UTC