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Fungrim entry: 42d832

θ2(r) ⁣(z,τ)=(πi)reπiτ/4n=(2n+1)rqn(n+1)w2n+1   where q=eπiτ,w=eπiz\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
Assumptions:zCandτHandrZ0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
TeX:
\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {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} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("42d832"),
    Formula(Equal(JacobiTheta(2, z, tau, r), Where(Mul(Mul(Pow(Mul(ConstPi, ConstI), r), Exp(Div(Mul(Mul(ConstPi, ConstI), tau), 4))), Sum(Mul(Mul(Pow(Add(Mul(2, n), 1), r), 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, r),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-20 18:07:53.062439 UTC