Fungrim entry: d530b1

\Lambda_{(a, b)} = \left\{ a m + b n : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

Assumptions:

a \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(\frac{b}{a}\right) \gt 0

TeX:

\Lambda_{(a, b)} = \left\{ a m + b n : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

a \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(\frac{b}{a}\right) \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Lattice`	$\Lambda_{(a, b)}$	Complex lattice with periods a, b
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part

Source code for this entry:

Entry(ID("d530b1"),
    Formula(Equal(Lattice(a, b), SetBuilder(Add(Mul(a, m), Mul(b, n)), Tuple(m, n), And(Element(m, ZZ), Element(n, ZZ))))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(CC, Set(0))), Element(b, SetMinus(CC, Set(0))), Greater(Im(Div(b, a)), 0))))

Topics using this entry

Weierstrass elliptic functions