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Fungrim entry: 7c4457

σ ⁣(z,τ)=z(m,n)Z2{(0,0)}(1zm+nτ)exp ⁣(zm+nτ+z22(m+nτ)2)\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:zCandτHandzΛ(1,τ)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
WeierstrassSigmaσ ⁣(z,τ)\sigma\!\left(z, \tau\right) Weierstrass sigma function
Productnf ⁣(n)\prod_{n} f\!\left(n\right) Product
Expez{e}^{z} Exponential function
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
LatticeΛ(a,b)\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)))))), Tuple(m, n), Element(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)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-16 21:17:18.797188 UTC