Fungrim entry: af0dfc

$\wp\!\left(z, \tau\right) = {\left(\pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}\right)}^{2} - \frac{{\pi}^{2}}{3} \left(\theta_{2}^{4}\!\left(0, \tau\right) + \theta_{3}^{4}\!\left(0, \tau\right)\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; z \notin \Lambda_{(1, \tau)}$
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$\wp\!\left(z, \tau\right)$ Weierstrass elliptic function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Lattice$\Lambda_{(a, b)}$ Complex lattice with periods a, b
Source code for this entry:
Entry(ID("af0dfc"),
Formula(Equal(WeierstrassP(z, tau), Sub(Pow(Mul(Mul(Mul(Pi, JacobiTheta(2, 0, tau)), JacobiTheta(3, 0, tau)), Div(JacobiTheta(4, z, tau), JacobiTheta(1, z, tau))), 2), Mul(Div(Pow(Pi, 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.

2021-03-15 19:12:00.328586 UTC