# Fungrim entry: 3e84e3

$\zeta\!\left(z, \tau\right) = \frac{1}{z} - \sum_{k=1}^{\infty} G_{2 k + 2}\!\left(\tau\right) {z}^{2 k + 1}$
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:
\zeta\!\left(z, \tau\right) = \frac{1}{z} - \sum_{k=1}^{\infty} G_{2 k + 2}\!\left(\tau\right) {z}^{2 k + 1}

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
WeierstrassZeta$\zeta\!\left(z, \tau\right)$ Weierstrass zeta function
Sum$\sum_{n} f(n)$ Sum
EisensteinG$G_{k}\!\left(\tau\right)$ Eisenstein series
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Abs$\left|z\right|$ Absolute value
Infimum$\mathop{\operatorname{inf}}\limits_{x \in S} f(x)$ Infimum of a set or function
Lattice$\Lambda_{(a, b)}$ Complex lattice with periods a, b
Source code for this entry:
Entry(ID("3e84e3"),
Formula(Equal(WeierstrassZeta(z, tau), Sub(Div(1, z), Sum(Mul(EisensteinG(Add(Mul(2, k), 2), tau), Pow(z, Add(Mul(2, k), 1))), For(k, 1, Infinity))))),
Variables(z, tau),
Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(z), Infimum(Set(Abs(s), ForElement(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.

2020-01-31 18:09:28.494564 UTC