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# Fungrim entry: 8f0a91

$R_F\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{\sin(\theta)}{\sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)}} \, d\theta \, d\phi$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)$
TeX:
R_F\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{\sin(\theta)}{\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)}} \, 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 \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonRF$R_F\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the first kind
Pi$\pi$ The constant pi (3.14...)
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Sin$\sin(z)$ Sine
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
Cos$\cos(z)$ Cosine
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("8f0a91"),
Formula(Equal(CarlsonRF(x, y, z), Mul(Div(1, Mul(4, Pi)), Integral(Integral(Div(Sin(theta), 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))))), 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), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))

## 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