Fungrim entry: 71a0ff

\operatorname{agm}\!\left(a, b\right) = \frac{\pi}{4} \frac{a + b}{K\!\left({\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{\pi}{4} \frac{a + b}{K\!\left({\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
`Pi`	$\pi$	The constant pi (3.14...)
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`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("71a0ff"),
    Formula(Equal(AGM(a, b), Mul(Div(Pi, 4), Div(Add(a, b), EllipticK(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)))))

Fungrim entry: 71a0ff

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