Fungrim home page

Fungrim entry: 45a130

θ4 ⁣(z+w,τ)θ4 ⁣(zw,τ)θ42 ⁣(0,τ)=θ32 ⁣(z,τ)θ32 ⁣(w,τ)θ22 ⁣(z,τ)θ22 ⁣(w,τ)=θ42 ⁣(z,τ)θ42 ⁣(w,τ)θ12 ⁣(z,τ)θ12 ⁣(w,τ)\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)
Assumptions:zCandwτandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, w \in \tau \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\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
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("45a130"),
    Formula(Equal(Mul(Mul(JacobiTheta(4, Add(z, w), tau), JacobiTheta(4, Sub(z, w), tau)), Pow(JacobiTheta(4, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-19 20:12:49.583742 UTC