Assumptions:
TeX:
\left(a_{n + 1}, b_{n + 1}\right) = \left(\frac{a_{n} + b_{n}}{2}, \sqrt{a_{n} b_{n}}\right)\; \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} \;\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 |
|---|---|---|
| Sqrt | Principal square root | |
| AGMSequence | Convergents in AGM iteration | |
| ZZGreaterEqual | Integers greater than or equal to n | |
| CC | Complex numbers | |
| Re | Real part | |
| Abs | Absolute value | |
| Arg | Complex argument | |
| Pi | The constant pi (3.14...) |
Source code for this entry:
Entry(ID("08b69d"),
Formula(Where(Equal(Tuple(a_(Add(n, 1)), b_(Add(n, 1))), Tuple(Div(Add(a_(n), b_(n)), 2), Sqrt(Mul(a_(n), b_(n))))), 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), 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)))))