Fungrim entry: d60119

\operatorname{agm}\!\left(c a, c b\right) = c \operatorname{agm}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; c = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

TeX:

\operatorname{agm}\!\left(c a, c b\right) = c \operatorname{agm}\!\left(a, b\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; c = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("d60119"),
    Formula(Equal(AGM(Mul(c, a), Mul(c, b)), Mul(c, AGM(a, b)))),
    Variables(a, b, c),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Or(Equal(a, 0), Equal(b, 0), Equal(c, 0), NotElement(Div(b, a), OpenClosedInterval(Neg(Infinity), 0))))))

Topics using this entry

Arithmetic-geometric mean