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Fungrim entry: 27b169

θ1 ⁣(4z,4τ)=θ1 ⁣(z,τ)θ1 ⁣(14z,τ)θ1 ⁣(14+z,τ)θ2 ⁣(z,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ3 ⁣(14,τ)\theta_{1}\!\left(4 z , 4 \tau\right) = \frac{\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(\frac{1}{4} - z , \tau\right) \theta_{1}\!\left(\frac{1}{4} + z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\theta_{1}\!\left(4 z , 4 \tau\right) = \frac{\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(\frac{1}{4} - z , \tau\right) \theta_{1}\!\left(\frac{1}{4} + z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \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(1, Mul(4, z), Mul(4, tau)), Div(Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(1, Sub(Div(1, 4), z), tau)), JacobiTheta(1, Add(Div(1, 4), z), tau)), JacobiTheta(2, z, tau)), Mul(Mul(JacobiTheta(3, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(3, Div(1, 4), 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.

2021-03-15 19:12:00.328586 UTC