Fungrim entry: b10ca7

\zeta\!\left(z, \tau\right) = \frac{1}{z} + \sum_{{m}^{2} + {n}^{2} \ne 0} \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_{{m}^{2} + {n}^{2} \ne 0} \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
`Pow`	${a}^{b}$	Power
`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), SumCondition(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))), Tuple(m, n), Unequal(Add(Pow(m, 2), Pow(n, 2)), 0))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

Topics using this entry

Weierstrass elliptic functions