# Fungrim entry: c7f885

$\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(\frac{a + b}{2}, \sqrt{a b}\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0\right) \;\mathbin{\operatorname{or}}\; \left|\arg(a)\right| + \left|\arg(b)\right| < \pi\right)$
TeX:
\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(\frac{a + b}{2}, \sqrt{a b}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0\right) \;\mathbin{\operatorname{or}}\; \left|\arg(a)\right| + \left|\arg(b)\right| < \pi\right)
Definitions:
Fungrim symbol Notation Short description
AGM$\operatorname{agm}\!\left(a, b\right)$ Arithmetic-geometric mean
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Abs$\left|z\right|$ Absolute value
Arg$\arg(z)$ Complex argument
Pi$\pi$ The constant pi (3.14...)
Source code for this entry:
Entry(ID("c7f885"),
Formula(Equal(AGM(a, b), AGM(Div(Add(a, b), 2), Sqrt(Mul(a, b))))),
Variables(a, b),
Assumptions(And(Element(a, CC), Element(b, CC), Or(Equal(a, 0), Equal(b, 0), And(Greater(Re(a), 0), Greater(Re(b), 0)), Less(Add(Abs(Arg(a)), Abs(Arg(b))), Pi)))))

## 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