Fungrim entry: 7c4457

\sigma\!\left(z, \tau\right) = z \prod_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \left(1 - \frac{z}{m + n \tau}\right) \exp\!\left(\frac{z}{m + n \tau} + \frac{{z}^{2}}{2 {\left(m + n \tau\right)}^{2}}\right)

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) = z \prod_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \left(1 - \frac{z}{m + n \tau}\right) \exp\!\left(\frac{z}{m + n \tau} + \frac{{z}^{2}}{2 {\left(m + n \tau\right)}^{2}}\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
`WeierstrassSigma`	$\sigma\!\left(z, \tau\right)$	Weierstrass sigma function
`Product`	$\prod_{n} f(n)$	Product
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`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("7c4457"),
    Formula(Equal(WeierstrassSigma(z, tau), Mul(z, Product(Mul(Sub(1, Div(z, Add(m, Mul(n, tau)))), Exp(Add(Div(z, Add(m, Mul(n, tau))), Div(Pow(z, 2), Mul(2, Pow(Add(m, Mul(n, tau)), 2)))))), ForElement(Tuple(m, n), SetMinus(Pow(ZZ, 2), Set(Tuple(0, 0)))))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

Topics using this entry

Weierstrass elliptic functions