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Fungrim entry: c7f885

agm ⁣(a,b)=agm ⁣(a+b2,ab)\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(\frac{a + b}{2}, \sqrt{a b}\right)
Assumptions:aC  and  bC  and  (a=0  or  b=0  or  (Re(a)>0  and  Re(b)>0)  or  arg(a)+arg(b)<π)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)
\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(\frac{a + b}{2}, \sqrt{a b}\right)

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)
Fungrim symbol Notation Short description
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Absz\left|z\right| Absolute value
Argarg(z)\arg(z) Complex argument
Piπ\pi The constant pi (3.14...)
Source code for this entry:
    Formula(Equal(AGM(a, b), AGM(Div(Add(a, b), 2), Sqrt(Mul(a, b))))),
    Variables(a, b),
    Assumptions(And(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)))))

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2021-03-15 19:12:00.328586 UTC