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Fungrim entry: e6d333

θ3 ⁣(z,τ2)=θ42 ⁣(z,τ)θ12 ⁣(z,τ)θ4 ⁣(0,τ2)\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \frac{\tau}{2}\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \frac{\tau}{2}\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
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("e6d333"),
    Formula(Equal(JacobiTheta(3, z, Div(tau, 2)), Div(Sub(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(1, z, tau), 2)), JacobiTheta(4, 0, Div(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.

2019-09-20 18:07:53.062439 UTC