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Fungrim entry: a255e1

θ2 ⁣(4z,4τ)=θ2 ⁣(18z,τ)θ2 ⁣(18+z,τ)θ2 ⁣(38z,τ)θ2 ⁣(38+z,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ3 ⁣(14,τ)\theta_{2}\!\left(4 z , 4 \tau\right) = \frac{\theta_{2}\!\left(\frac{1}{8} - z , \tau\right) \theta_{2}\!\left(\frac{1}{8} + z , \tau\right) \theta_{2}\!\left(\frac{3}{8} - z , \tau\right) \theta_{2}\!\left(\frac{3}{8} + 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}
TeX:
\theta_{2}\!\left(4 z , 4 \tau\right) = \frac{\theta_{2}\!\left(\frac{1}{8} - z , \tau\right) \theta_{2}\!\left(\frac{1}{8} + z , \tau\right) \theta_{2}\!\left(\frac{3}{8} - z , \tau\right) \theta_{2}\!\left(\frac{3}{8} + 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}
Definitions:
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:
Entry(ID("a255e1"),
    Formula(Equal(JacobiTheta(2, Mul(4, z), Mul(4, tau)), Div(Mul(Mul(Mul(JacobiTheta(2, Sub(Div(1, 8), z), tau), JacobiTheta(2, Add(Div(1, 8), z), tau)), JacobiTheta(2, Sub(Div(3, 8), z), tau)), JacobiTheta(2, Add(Div(3, 8), 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