Fungrim entry: 417619

\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 \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)

TeX:

\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)

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("417619"),
    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)))))

Topics using this entry

Arithmetic-geometric mean