Fungrim home page

Fungrim entry: 75e692

agm ⁣(1,z)ananbn   where (an,bn)=agmn ⁣(1,z)\left|\operatorname{agm}\!\left(1, z\right) - a_{n}\right| \le \left|a_{n} - b_{n}\right|\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, z\right)
Assumptions:zC  and  Re(z)0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0
\left|\operatorname{agm}\!\left(1, z\right) - a_{n}\right| \le \left|a_{n} - b_{n}\right|\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
AGMSequenceagmn ⁣(a,b)\operatorname{agm}_{n}\!\left(a, b\right) Convergents in AGM iteration
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    Formula(Where(LessEqual(Abs(Sub(AGM(1, z), a_(n))), Abs(Sub(a_(n), b_(n)))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, z)))),
    Assumptions(And(Element(z, CC), GreaterEqual(Re(z), 0))))

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