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Fungrim entry: 9973ef

θ2 ⁣(0,τ)θ4 ⁣(0,τ)θ2 ⁣(z+w,τ)θ4 ⁣(zw,τ)=θ2 ⁣(z,τ)θ4 ⁣(z,τ)θ2 ⁣(w,τ)θ4 ⁣(w,τ)θ1 ⁣(z,τ)θ3 ⁣(z,τ)θ1 ⁣(w,τ)θ3 ⁣(w,τ)\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)
Assumptions:zCandwτandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, w \in \tau \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, w \in \tau \,\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("9973ef"),
    Formula(Equal(Mul(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(2, Add(z, w), tau)), JacobiTheta(4, Sub(z, w), tau)), Sub(Mul(Mul(Mul(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(2, w, tau)), JacobiTheta(4, w, tau)), Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(3, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(3, w, tau))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), 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.

2019-10-05 13:11:19.856591 UTC