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Fungrim entry: 700d94

θ1 ⁣(z,τ)=ieπiτ/4n=(1)nqn(n+1)w2n+1   where q=eπiτ,w=eπiz\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \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_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \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
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
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:
Entry(ID("700d94"),
    Formula(Equal(JacobiTheta(1, z, tau), Where(Mul(Mul(Neg(ConstI), Exp(Div(Mul(Mul(ConstPi, ConstI), tau), 4))), Sum(Mul(Mul(Pow(-1, n), Pow(q, Mul(n, Add(n, 1)))), Pow(w, Add(Mul(2, n), 1))), Tuple(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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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC