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Fungrim entry: 21dc98

θ4 ⁣(2z,τ)=θ12 ⁣(z,τ)θ22 ⁣(z,τ)+θ32 ⁣(z,τ)θ42 ⁣(z,τ)θ4 ⁣(0,τ)θ32 ⁣(0,τ)\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(4, Mul(2, z), tau), Div(Add(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(2, z, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2))), Mul(JacobiTheta(4, 0, tau), Pow(JacobiTheta(3, 0, tau), 2))))),
    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.

2021-03-15 19:12:00.328586 UTC