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Fungrim entry: 95fb3e

agm ⁣(a,b)=limnan=limnbn   where (an,bn)=agmn ⁣(a,b)\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:aC  and  bCa \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
\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}
Fungrim symbol Notation Short description
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
SequenceLimitlimnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
Infinity\infty Positive infinity
AGMSequenceagmn ⁣(a,b)\operatorname{agm}_{n}\!\left(a, b\right) Convergents in AGM iteration
CCC\mathbb{C} Complex numbers
Source code for this entry:
    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))))

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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