# 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

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