# Fungrim entry: 08b69d

$\left(a_{n + 1}, b_{n + 1}\right) = \left(\frac{a_{n} + b_{n}}{2}, \sqrt{a_{n} b_{n}}\right)\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0\right) \;\mathbin{\operatorname{or}}\; \left|\arg(a)\right| + \left|\arg(b)\right| < \pi\right)$
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$\sqrt{z}$ Principal square root
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
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("08b69d"),
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)))))