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Fungrim entry: 0a9ec2

θ3 ⁣(2z,2τ)=θ3 ⁣(14z,τ)θ3 ⁣(14+z,τ)θ4 ⁣(0,2τ)\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}\!\left(\frac{1}{4} - z , \tau\right) \theta_{3}\!\left(\frac{1}{4} + z , \tau\right)}{\theta_{4}\!\left(0 , 2 \tau\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}\!\left(\frac{1}{4} - z , \tau\right) \theta_{3}\!\left(\frac{1}{4} + z , \tau\right)}{\theta_{4}\!\left(0 , 2 \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
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(3, Mul(2, z), Mul(2, tau)), Div(Mul(JacobiTheta(3, Sub(Div(1, 4), z), tau), JacobiTheta(3, Add(Div(1, 4), z), tau)), JacobiTheta(4, 0, Mul(2, tau))))),
    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.

2020-01-31 18:09:28.494564 UTC