Fungrim entry: 95fb3e

\operatorname{agm}\!\left(a, b\right) = \lim_{n \to \infty} a_{n} = \lim_{n \to \infty} b_{n}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, b\right) = \lim_{n \to \infty} a_{n} = \lim_{n \to \infty} b_{n}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(a, b\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Infinity`	$\infty$	Positive infinity
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("95fb3e"),
    Formula(Where(Equal(AGM(a, b), SequenceLimit(a_(n), For(n, Infinity)), SequenceLimit(b_(n), For(n, Infinity))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

Topics using this entry

Arithmetic-geometric mean