Fungrim home page

Fungrim entry: 3e84e3

ζ ⁣(z,τ)=1zk=1G2k+2 ⁣(τ)z2k+1\zeta\!\left(z, \tau\right) = \frac{1}{z} - \sum_{k=1}^{\infty} G_{2 k + 2}\!\left(\tau\right) {z}^{2 k + 1}
Assumptions:zC  and  τH  and  z<inf{s:sΛ(1,τ){0}}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\}
\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\}
Fungrim symbol Notation Short description
WeierstrassZetaζ ⁣(z,τ)\zeta\!\left(z, \tau\right) Weierstrass zeta function
Sumnf(n)\sum_{n} f(n) Sum
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Absz\left|z\right| Absolute value
InfimuminfxSf(x)\mathop{\operatorname{inf}}\limits_{x \in S} f(x) Infimum of a set or function
LatticeΛ(a,b)\Lambda_{(a, b)} Complex lattice with periods a, b
Source code for this entry:
    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.

2021-03-15 19:12:00.328586 UTC