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

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