# 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

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