Fungrim entry: fa6ff7

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(x, s y\right)\; \text{ where } x = \frac{a + b}{2},\;y = \sqrt{a b},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(x, s y\right)\; \text{ where } x = \frac{a + b}{2},\;y = \sqrt{a b},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Re`	$\operatorname{Re}(z)$	Real part
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fa6ff7"),
    Formula(Equal(AGM(a, b), Where(AGM(x, Mul(s, y)), Def(x, Div(Add(a, b), 2)), Def(y, Sqrt(Mul(a, b))), Def(s, Cases(Tuple(Pos(1), Or(Equal(y, 0), GreaterEqual(Re(Div(x, y)), 0))), Tuple(Neg(1), Otherwise)))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

Topics using this entry

Arithmetic-geometric mean