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Fungrim entry: d6d836

agm ⁣(a,b)=a+b22F1 ⁣(12,12,1,(aba+b)2)\operatorname{agm}\!\left(a, b\right) = \frac{a + b}{2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, {\left(\frac{a - b}{a + b}\right)}^{2}\right)}
Assumptions:aC  and  bC  and  b0  and  ab(,0]a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]
\operatorname{agm}\!\left(a, b\right) = \frac{a + b}{2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, {\left(\frac{a - b}{a + b}\right)}^{2}\right)}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]
Fungrim symbol Notation Short description
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Equal(AGM(a, b), Div(Add(a, b), Mul(2, Hypergeometric2F1(Div(1, 2), Div(1, 2), 1, Pow(Div(Sub(a, b), Add(a, b)), 2)))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC