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Fungrim entry: 58d67b

 ⁣(z,τ)=1z2+(m,n)Z2{(0,0)}1(z+m+nτ)21(m+nτ)2\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{{\left(z + m + n \tau\right)}^{2}} - \frac{1}{{\left(m + n \tau\right)}^{2}}
Assumptions:zC  and  τH  and  zΛ(1,τ)z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; z \notin \Lambda_{(1, \tau)}
\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{{\left(z + m + n \tau\right)}^{2}} - \frac{1}{{\left(m + n \tau\right)}^{2}}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; z \notin \Lambda_{(1, \tau)}
Fungrim symbol Notation Short description
WeierstrassP ⁣(z,τ)\wp\!\left(z, \tau\right) Weierstrass elliptic function
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
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:
    Formula(Equal(WeierstrassP(z, tau), Add(Div(1, Pow(z, 2)), Sum(Sub(Div(1, Pow(Add(Add(z, m), Mul(n, tau)), 2)), Div(1, 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)))))

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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