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

Arithmetic-geometric mean