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Fungrim entry: 06633e

θ2 ⁣(z,τ)=2n=0eπi(n+1/2)2τcos ⁣((2n+1)πz)\theta_2\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \cos\!\left(\left(2 n + 1\right) \pi z\right)
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_2\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \cos\!\left(\left(2 n + 1\right) \pi z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta2θ2 ⁣(z,τ)\theta_2\!\left(z, \tau\right) Jacobi theta function
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("06633e"),
    Formula(Equal(JacobiTheta2(z, tau), Mul(2, Sum(Mul(Exp(Mul(Mul(Mul(ConstPi, ConstI), Pow(Add(n, Div(1, 2)), 2)), tau)), Cos(Mul(Mul(Add(Mul(2, n), 1), ConstPi), z))), Tuple(n, 0, 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.

2019-06-18 07:49:59.356594 UTC