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Fungrim entry: 8a34d1

θ4 ⁣(z,τ)=1+2n=1(1)nqn2cos ⁣(2nπz)   where q=eπiτ\theta_{4}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{4}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

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
ConstPiπ\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:
Entry(ID("8a34d1"),
    Formula(Equal(JacobiTheta(4, z, tau), Where(Add(1, Mul(2, Sum(Mul(Mul(Pow(-1, n), Pow(q, Pow(n, 2))), Cos(Mul(Mul(Mul(2, n), ConstPi), z))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2019-10-05 13:11:19.856591 UTC