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Fungrim entry: af0dfc

 ⁣(z,τ)=(πθ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(z,τ)θ1 ⁣(z,τ))2π23(θ24 ⁣(0,τ)+θ34 ⁣(0,τ))\wp\!\left(z, \tau\right) = {\left(\pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}\right)}^{2} - \frac{{\pi}^{2}}{3} \left(\theta_{2}^{4}\!\left(0, \tau\right) + \theta_{3}^{4}\!\left(0, \tau\right)\right)
Assumptions:zCandτHandzΛ(1,τ)z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, z \notin \Lambda_{(1, \tau)}
TeX:
\wp\!\left(z, \tau\right) = {\left(\pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}\right)}^{2} - \frac{{\pi}^{2}}{3} \left(\theta_{2}^{4}\!\left(0, \tau\right) + \theta_{3}^{4}\!\left(0, \tau\right)\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, z \notin \Lambda_{(1, \tau)}
Definitions:
Fungrim symbol Notation Short description
WeierstrassP ⁣(z,τ)\wp\!\left(z, \tau\right) Weierstrass elliptic function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
LatticeΛ(a,b)\Lambda_{(a, b)} Complex lattice with periods a, b
Source code for this entry:
Entry(ID("af0dfc"),
    Formula(Equal(WeierstrassP(z, tau), Sub(Pow(Mul(Mul(Mul(ConstPi, JacobiTheta(2, 0, tau)), JacobiTheta(3, 0, tau)), Div(JacobiTheta(4, z, tau), JacobiTheta(1, z, tau))), 2), Mul(Div(Pow(ConstPi, 2), 3), Add(Pow(JacobiTheta(2, 0, tau), 4), Pow(JacobiTheta(3, 0, tau), 4)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

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

2019-09-16 21:17:18.797188 UTC