$\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{k=1}^{\infty} \left(2 k + 1\right) G_{2 k + 2}\!\left(\tau\right) {z}^{2 k}$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left|z\right| < \operatorname{inf} \left\{ \left|s\right| : s \in \Lambda_{(1, \tau)} \setminus \left\{0\right\} \right\}$
TeX:
\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{k=1}^{\infty} \left(2 k + 1\right) G_{2 k + 2}\!\left(\tau\right) {z}^{2 k}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left|z\right| < \operatorname{inf} \left\{ \left|s\right| : s \in \Lambda_{(1, \tau)} \setminus \left\{0\right\} \right\}
Definitions:
Fungrim symbol Notation Short description
WeierstrassP$\wp\!\left(z, \tau\right)$ Weierstrass elliptic function
Pow${a}^{b}$ Power
Sum$\sum_{n} f\!\left(n\right)$ Sum
EisensteinG$G_{k}\!\left(\tau\right)$ Eisenstein series
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Abs$\left|z\right|$ Absolute value
Infimum$\mathop{\operatorname{inf}}\limits_{P\left(x\right)} f\!\left(x\right)$ Infimum of a set or function
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
Lattice$\Lambda_{(a, b)}$ Complex lattice with periods a, b
Source code for this entry:
Entry(ID("9bf0ad"),
Formula(Equal(WeierstrassP(z, tau), Add(Div(1, Pow(z, 2)), Sum(Mul(Mul(Add(Mul(2, k), 1), EisensteinG(Add(Mul(2, k), 2), tau)), Pow(z, Mul(2, k))), Tuple(k, 1, Infinity))))),
Variables(z, tau),
Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(z), Infimum(SetBuilder(Abs(s), s, Element(s, SetMinus(Lattice(1, tau), Set(0)))))))))

## Topics using this entry

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