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)))))))))

Fungrim entry: 3e84e3

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