# Fungrim entry: 75e692

$\left|\operatorname{agm}\!\left(1, z\right) - a_{n}\right| \le \left|a_{n} - b_{n}\right|\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, z\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0$
TeX:
\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
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
AGM$\operatorname{agm}\!\left(a, b\right)$ Arithmetic-geometric mean
AGMSequence$\operatorname{agm}_{n}\!\left(a, b\right)$ Convergents in AGM iteration
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("75e692"),
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)))),
Variables(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