Fungrim entry: a2b0f9

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

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

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

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`Re`	$\operatorname{Re}(z)$	Real part
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a2b0f9"),
    Formula(Where(Equal(Tuple(a_(Add(n, 1)), b_(Add(n, 1))), Where(Tuple(x, Mul(s, y)), Def(x, Div(Add(a_(n), b_(n)), 2)), Def(y, Sqrt(Mul(a_(n), b_(n)))), Def(s, Cases(Tuple(Pos(1), Or(Equal(y, 0), GreaterEqual(Re(Div(x, y)), 0))), Tuple(Neg(1), Otherwise))))), Def(Tuple(a_(k), b_(k)), AGMSequence(k, a, b)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC))))

Topics using this entry

Arithmetic-geometric mean