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Fungrim entry: 2ae142

θ1(r) ⁣(z,τ)=i(πi)reπiτ/4n=(1)n(2n+1)rqn(n+1)w2n+1   where q=eπiτ,  w=eπiz\theta^{(r)}_{1}\!\left(z , \tau\right) = -i {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {\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:zC  and  τH  and  rZ0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
\theta^{(r)}_{1}\!\left(z , \tau\right) = -i {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {\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}
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
Sumnf(n)\sum_{n} f(n) 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:
    Formula(Equal(JacobiTheta(1, z, tau, r), Where(Mul(Mul(Mul(Neg(ConstI), Pow(Mul(Pi, ConstI), r)), Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Mul(Mul(Pow(-1, n), Pow(Add(Mul(2, n), 1), r)), Pow(q, Mul(n, Add(n, 1)))), Pow(w, Add(Mul(2, n), 1))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau, r),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

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2021-03-15 19:12:00.328586 UTC