Fungrim entry: 31a3ba

R_D\!\left(x, y, z\right) = 2 R_D\!\left(x + \lambda, y + \lambda, z + \lambda\right) + \frac{3}{\sqrt{z} \left(z + \lambda\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}}\; z \ne 0

TeX:

R_D\!\left(x, y, z\right) = 2 R_D\!\left(x + \lambda, y + \lambda, z + \lambda\right) + \frac{3}{\sqrt{z} \left(z + \lambda\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}}\; z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`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("31a3ba"),
    Formula(Equal(CarlsonRD(x, y, z), Where(Add(Mul(2, CarlsonRD(Add(x, lamda), Add(y, lamda), Add(z, lamda))), Div(3, Mul(Sqrt(z), Add(z, lamda)))), 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), NotEqual(z, 0))))

Topics using this entry

Carlson symmetric elliptic integrals