Fungrim entry: 35403b

\sigma\!\left(z + 1, \tau\right) = -\exp\!\left(2 \left(z + \frac{1}{2}\right) \zeta\!\left(\frac{1}{2}, \tau\right)\right) \sigma\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\sigma\!\left(z + 1, \tau\right) = -\exp\!\left(2 \left(z + \frac{1}{2}\right) \zeta\!\left(\frac{1}{2}, \tau\right)\right) \sigma\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`WeierstrassSigma`	$\sigma\!\left(z, \tau\right)$	Weierstrass sigma function
`Exp`	${e}^{z}$	Exponential function
`WeierstrassZeta`	$\zeta\!\left(z, \tau\right)$	Weierstrass zeta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("35403b"),
    Formula(Equal(WeierstrassSigma(Add(z, 1), tau), Neg(Mul(Exp(Mul(Mul(2, Add(z, Div(1, 2))), WeierstrassZeta(Div(1, 2), tau))), WeierstrassSigma(z, tau))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Weierstrass elliptic functions