Fungrim entry: 41f67b

2 a_{n + 1} = a_{n} + b_{n} \;\mathbin{\operatorname{and}}\; b_{n + 1}^{2} = a_{n} b_{n}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

2 a_{n + 1} = a_{n} + b_{n} \;\mathbin{\operatorname{and}}\; b_{n + 1}^{2} = a_{n} b_{n}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("41f67b"),
    Formula(Where(And(Equal(Mul(2, a_(Add(n, 1))), Add(a_(n), b_(n))), Equal(Pow(b_(Add(n, 1)), 2), Mul(a_(n), b_(n)))), Def(Tuple(a_(k), b_(k)), AGMSequence(k, a, b)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC))))

Topics using this entry

Arithmetic-geometric mean