Fungrim entry: 0d8639

R_G\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)} \sin(\theta) \, d\theta \, d\phi

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0

TeX:

R_G\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)} \sin(\theta) \, d\theta \, d\phi

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`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
`Sin`	$\sin(z)$	Sine
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("0d8639"),
    Formula(Equal(CarlsonRG(x, y, z), Mul(Div(1, Mul(4, Pi)), Integral(Integral(Mul(Sqrt(Add(Add(Mul(Mul(x, Pow(Sin(theta), 2)), Pow(Cos(phi), 2)), Mul(Mul(y, Pow(Sin(theta), 2)), Pow(Sin(phi), 2))), Mul(z, Pow(Cos(theta), 2)))), Sin(theta)), For(theta, 0, Pi)), For(phi, 0, Mul(2, Pi)))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), GreaterEqual(Re(x), 0), GreaterEqual(Re(y), 0), GreaterEqual(Re(z), 0))))

Topics using this entry

Carlson symmetric elliptic integrals