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

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