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

Arithmetic-geometric mean