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Fungrim entry: 34d1c6

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \tau \;\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(Mul(Mul(Mul(JacobiTheta(3, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(1, Add(z, w), tau)), JacobiTheta(2, Sub(z, w), tau)), Add(Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), JacobiTheta(3, w, tau)), JacobiTheta(4, w, tau)), Mul(Mul(Mul(JacobiTheta(3, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(2, 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.

2021-03-15 19:12:00.328586 UTC