# Fungrim entry: d6d836

$\operatorname{agm}\!\left(a, b\right) = \frac{a + b}{2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, {\left(\frac{a - b}{a + b}\right)}^{2}\right)}$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]$
TeX:
\operatorname{agm}\!\left(a, b\right) = \frac{a + b}{2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, {\left(\frac{a - b}{a + b}\right)}^{2}\right)}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]
Definitions:
Fungrim symbol Notation Short description
AGM$\operatorname{agm}\!\left(a, b\right)$ Arithmetic-geometric mean
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("d6d836"),
Formula(Equal(AGM(a, b), Div(Add(a, b), Mul(2, Hypergeometric2F1(Div(1, 2), Div(1, 2), 1, Pow(Div(Sub(a, b), Add(a, b)), 2)))))),
Variables(a, b),
Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 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