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({\left(\theta_2\!\left(0, \tau\right)\right)}^{4} + {\left(\theta_3\!\left(0, \tau\right)\right)}^{4}\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({\left(\theta_2\!\left(0, \tau\right)\right)}^{4} + {\left(\theta_3\!\left(0, \tau\right)\right)}^{4}\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
`ConstPi`	$\pi$	The constant pi (3.14...)
`JacobiTheta2`	$\theta_2\!\left(z, \tau\right)$	Jacobi theta function
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	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(ConstPi, JacobiTheta2(0, tau)), JacobiTheta3(0, tau)), Div(JacobiTheta4(z, tau), JacobiTheta1(z, tau))), 2), Mul(Div(Pow(ConstPi, 2), 3), Add(Pow(JacobiTheta2(0, tau), 4), Pow(JacobiTheta3(0, tau), 4)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

Topics using this entry

Weierstrass elliptic functions