Fungrim entry: 8e6189

R_F\!\left(x, y, z\right) = R_F\!\left(\frac{x + \lambda}{4}, \frac{y + \lambda}{4}, \frac{z + \lambda}{4}\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}

TeX:

R_F\!\left(x, y, z\right) = R_F\!\left(\frac{x + \lambda}{4}, \frac{y + \lambda}{4}, \frac{z + \lambda}{4}\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}

Definitions:

Fungrim symbol	Notation	Short description
`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("8e6189"),
    Formula(Equal(CarlsonRF(x, y, z), Where(CarlsonRF(Div(Add(x, lamda), 4), Div(Add(y, lamda), 4), Div(Add(z, lamda), 4)), 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))))

Topics using this entry

Carlson symmetric elliptic integrals