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Fungrim entry: 580ba0

θ3 ⁣(z,τ)=n=eπi(n2τ+2nz)\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

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\!\left(n\right) Sum
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
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("580ba0"),
    Formula(Equal(JacobiTheta(3, z, tau), Sum(Exp(Mul(Mul(ConstPi, ConstI), Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z)))), Tuple(n, Neg(Infinity), Infinity)))),
    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.

2019-08-19 14:38:23.809000 UTC