Fungrim entry: 47cf5d

R_G\!\left(x, y, z\right) = 2 R_G\!\left(x + \lambda, y + \lambda, z + \lambda\right) - \frac{1}{2} \left(\lambda R_F\!\left(x, y, z\right) + \sqrt{x} + \sqrt{y} + \sqrt{z}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \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(\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_G\!\left(x, y, z\right) = 2 R_G\!\left(x + \lambda, y + \lambda, z + \lambda\right) - \frac{1}{2} \left(\lambda R_F\!\left(x, y, z\right) + \sqrt{x} + \sqrt{y} + \sqrt{z}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\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
`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
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("47cf5d"),
    Formula(Equal(CarlsonRG(x, y, z), Where(Sub(Mul(2, CarlsonRG(Add(x, lamda), Add(y, lamda), Add(z, lamda))), Mul(Div(1, 2), Add(Add(Add(Mul(lamda, CarlsonRF(x, y, z)), Sqrt(x)), Sqrt(y)), Sqrt(z)))), Def(lamda, Add(Add(Mul(Sqrt(x), Sqrt(y)), Mul(Sqrt(y), Sqrt(z))), Mul(Sqrt(x), Sqrt(z))))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), 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

Carlson symmetric elliptic integrals