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# Fungrim entry: fa6ff7

$\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(x, s y\right)\; \text{ where } x = \frac{a + b}{2},\;y = \sqrt{a b},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}$
TeX:
\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(x, s y\right)\; \text{ where } x = \frac{a + b}{2},\;y = \sqrt{a b},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
AGM$\operatorname{agm}\!\left(a, b\right)$ Arithmetic-geometric mean
Sqrt$\sqrt{z}$ Principal square root
Re$\operatorname{Re}(z)$ Real part
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("fa6ff7"),
Formula(Equal(AGM(a, b), Where(AGM(x, Mul(s, y)), Def(x, Div(Add(a, b), 2)), Def(y, Sqrt(Mul(a, b))), Def(s, Cases(Tuple(Pos(1), Or(Equal(y, 0), GreaterEqual(Re(Div(x, y)), 0))), Tuple(Neg(1), Otherwise)))))),
Variables(a, b),
Assumptions(And(Element(a, CC), Element(b, CC))))

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2021-03-15 19:12:00.328586 UTC