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Fungrim entry: 06633e

θ2 ⁣(z,τ)=2eπiτ/4n=0qn(n+1)cos ⁣((2n+1)πz)   where q=eπiτ\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {q}^{n \left(n + 1\right)} \cos\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
TeX:
\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {q}^{n \left(n + 1\right)} \cos\!\left(\left(2 n + 1\right) \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
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Coscos(z)\cos(z) Cosine
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("06633e"),
    Formula(Equal(JacobiTheta(2, z, tau), Where(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Pow(q, Mul(n, Add(n, 1))), Cos(Mul(Mul(Add(Mul(2, n), 1), Pi), z))), For(n, 0, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2020-04-08 16:14:44.404316 UTC