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Fungrim entry: 9b7d8c

θ22 ⁣(0,τ)=n=1cos ⁣(πτ(n+12))\theta_{2}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \tau \left(n + \frac{1}{2}\right)\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{2}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \tau \left(n + \frac{1}{2}\right)\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
ConstPiπ\pi The constant pi (3.14...)
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("9b7d8c"),
    Formula(Equal(Pow(JacobiTheta(2, 0, tau), 2), Sum(Div(1, Cos(Mul(Mul(ConstPi, tau), Add(n, Div(1, 2))))), Tuple(n, Neg(Infinity), Infinity)))),
    Variables(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.

2019-09-20 18:07:53.062439 UTC