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# Fungrim entry: b10ca7

$\zeta\!\left(z, \tau\right) = \frac{1}{z} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{z - m - n \tau} + \frac{1}{m + n \tau} + \frac{z}{{\left(m + n \tau\right)}^{2}}$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, z \notin \Lambda_{(1, \tau)}$
TeX:
\zeta\!\left(z, \tau\right) = \frac{1}{z} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{z - m - n \tau} + \frac{1}{m + n \tau} + \frac{z}{{\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)}
Definitions:
Fungrim symbol Notation Short description
WeierstrassZeta$\zeta\!\left(z, \tau\right)$ Weierstrass zeta function
Sum$\sum_{n} f(n)$ Sum
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("b10ca7"),
Formula(Equal(WeierstrassZeta(z, tau), Add(Div(1, z), Sum(Add(Add(Div(1, Sub(Sub(z, m), Mul(n, tau))), Div(1, Add(m, Mul(n, tau)))), Div(z, 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.

2020-01-31 18:09:28.494564 UTC