Fungrim entry: b96c9d

\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}\right) \frac{\theta_1\!\left(z, \tau\right)}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, z \notin \Lambda_{(1, \tau)}

TeX:

\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}\right) \frac{\theta_1\!\left(z, \tau\right)}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}

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
`WeierstrassSigma`	$\sigma\!\left(z, \tau\right)$	Weierstrass sigma function
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`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("b96c9d"),
    Formula(Equal(WeierstrassSigma(z, tau), Mul(Exp(Mul(Neg(Div(Pow(z, 2), 6)), Div(Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 3)), Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 1))))), Div(JacobiTheta1(z, tau), Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 1)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

Topics using this entry

Weierstrass elliptic functions