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Fungrim entry: 0d8639

RG ⁣(x,y,z)=14π02π0πxsin2 ⁣(θ)cos2 ⁣(ϕ)+ysin2 ⁣(θ)sin2 ⁣(ϕ)+zcos2 ⁣(θ)sin(θ)dθdϕR_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:xC  and  yC  and  zC  and  Re(x)0  and  Re(y)0  and  Re(z)0x \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
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
Fungrim symbol Notation Short description
CarlsonRGRG ⁣(x,y,z)R_G\!\left(x, y, z\right) Carlson symmetric elliptic integral of the second kind
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
Sinsin(z)\sin(z) Sine
Coscos(z)\cos(z) Cosine
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    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))))

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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