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Fungrim entry: fa6ff7

agm ⁣(a,b)=agm ⁣(x,sy)   where x=a+b2,  y=ab,  s={+1,y=0  or  Re ⁣(xy)01,otherwise\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:aC  and  bCa \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
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Sqrtz\sqrt{z} Principal square root
ReRe(z)\operatorname{Re}(z) Real part
CCC\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