# Fungrim entry: 7609c8

$2 R_G\!\left(x, y, z\right) = z R_F\!\left(x, y, z\right) - \frac{\left(x - z\right) \left(y - z\right)}{3} R_D\!\left(x, y, z\right) + \frac{\sqrt{x} \sqrt{y}}{\sqrt{z}}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right) \;\mathbin{\operatorname{and}}\; z \ne 0$
TeX:
2 R_G\!\left(x, y, z\right) = z R_F\!\left(x, y, z\right) - \frac{\left(x - z\right) \left(y - z\right)}{3} R_D\!\left(x, y, z\right) + \frac{\sqrt{x} \sqrt{y}}{\sqrt{z}}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right) \;\mathbin{\operatorname{and}}\; z \ne 0
Definitions:
Fungrim symbol Notation Short description
CarlsonRG$R_G\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the second kind
CarlsonRF$R_F\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the first kind
CarlsonRD$R_D\!\left(x, y, z\right)$ Degenerate Carlson symmetric elliptic integral of the third kind
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("7609c8"),
Formula(Equal(Mul(2, CarlsonRG(x, y, z)), Add(Sub(Mul(z, CarlsonRF(x, y, z)), Mul(Div(Mul(Sub(x, z), Sub(y, z)), 3), CarlsonRD(x, y, z))), Div(Mul(Sqrt(x), Sqrt(y)), Sqrt(z))))),
Variables(x, y, z),
Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Or(NotEqual(x, 0), 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