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Fungrim entry: 417619

agm ⁣(a,b)=π2I   where I=0π/21a2cos2 ⁣(x)+b2sin2 ⁣(x)dx\operatorname{agm}\!\left(a, b\right) = \frac{\pi}{2 I}\; \text{ where } I = \int_{0}^{\pi / 2} \frac{1}{\sqrt{{a}^{2} \cos^{2}\!\left(x\right) + {b}^{2} \sin^{2}\!\left(x\right)}} \, dx
Assumptions:a(0,)  and  b(0,)a \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)
\operatorname{agm}\!\left(a, b\right) = \frac{\pi}{2 I}\; \text{ where } I = \int_{0}^{\pi / 2} \frac{1}{\sqrt{{a}^{2} \cos^{2}\!\left(x\right) + {b}^{2} \sin^{2}\!\left(x\right)}} \, dx

a \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)
Fungrim symbol Notation Short description
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Piπ\pi The constant pi (3.14...)
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
Coscos(z)\cos(z) Cosine
Sinsin(z)\sin(z) Sine
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Equal(AGM(a, b), Where(Div(Pi, Mul(2, I)), Def(I, Integral(Div(1, Sqrt(Add(Mul(Pow(a, 2), Pow(Cos(x), 2)), Mul(Pow(b, 2), Pow(Sin(x), 2))))), For(x, 0, Div(Pi, 2))))))),
    Variables(a, b),
    Assumptions(And(Element(a, OpenInterval(0, Infinity)), Element(b, OpenInterval(0, Infinity)))))

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2021-03-15 19:12:00.328586 UTC