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Fungrim entry: 64081c

θ2 ⁣(z,τ)=θ2 ⁣(0,τ)cos ⁣(πz)n=1cos ⁣(π(nτ+z))cos ⁣(π(nτz))cos2 ⁣(πnτ)\theta_{2}\!\left(z , \tau\right) = \theta_{2}\!\left(0 , \tau\right) \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(n \tau + z\right)\right) \cos\!\left(\pi \left(n \tau - z\right)\right)}{\cos^{2}\!\left(\pi n \tau\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{2}\!\left(z , \tau\right) = \theta_{2}\!\left(0 , \tau\right) \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(n \tau + z\right)\right) \cos\!\left(\pi \left(n \tau - z\right)\right)}{\cos^{2}\!\left(\pi n \tau\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
ConstPiπ\pi The constant pi (3.14...)
Productnf ⁣(n)\prod_{n} f\!\left(n\right) Product
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("64081c"),
    Formula(Equal(JacobiTheta(2, z, tau), Mul(Mul(JacobiTheta(2, 0, tau), Cos(Mul(ConstPi, z))), Product(Div(Mul(Cos(Mul(ConstPi, Add(Mul(n, tau), z))), Cos(Mul(ConstPi, Sub(Mul(n, tau), z)))), Pow(Cos(Mul(Mul(ConstPi, n), tau)), 2)), For(n, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2019-09-22 15:43:45.410764 UTC