Fungrim entry: 791c44

R_J\!\left(x, y, z, w\right) = 2 R_J\!\left(x + \lambda, y + \lambda, z + \lambda, w + \lambda\right) + \frac{6}{d} R_C\!\left(1, 1 + \frac{\delta}{{d}^{2}}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z},\;\delta = \left(w - x\right) \left(w - y\right) \left(w - z\right),\;d = \left(\sqrt{w} + \sqrt{x}\right) \left(\sqrt{w} + \sqrt{y}\right) \left(\sqrt{w} + \sqrt{z}\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \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}}\; \operatorname{Re}(w) > 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_J\!\left(x, y, z, w\right) = 2 R_J\!\left(x + \lambda, y + \lambda, z + \lambda, w + \lambda\right) + \frac{6}{d} R_C\!\left(1, 1 + \frac{\delta}{{d}^{2}}\right)\; \text{ where } \lambda = \sqrt{x} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{x} \sqrt{z},\;\delta = \left(w - x\right) \left(w - y\right) \left(w - z\right),\;d = \left(\sqrt{w} + \sqrt{x}\right) \left(\sqrt{w} + \sqrt{y}\right) \left(\sqrt{w} + \sqrt{z}\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \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}}\; \operatorname{Re}(w) > 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
`CarlsonRJ`	$R_J\!\left(x, y, z, w\right)$	Carlson symmetric elliptic integral of the third kind
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("791c44"),
    Formula(Equal(CarlsonRJ(x, y, z, w), Where(Add(Mul(2, CarlsonRJ(Add(x, lamda), Add(y, lamda), Add(z, lamda), Add(w, lamda))), Mul(Div(6, d), CarlsonRC(1, Add(1, Div(delta, Pow(d, 2)))))), Def(lamda, Add(Add(Mul(Sqrt(x), Sqrt(y)), Mul(Sqrt(y), Sqrt(z))), Mul(Sqrt(x), Sqrt(z)))), Def(delta, Mul(Mul(Sub(w, x), Sub(w, y)), Sub(w, z))), Def(d, Mul(Mul(Add(Sqrt(w), Sqrt(x)), Add(Sqrt(w), Sqrt(y))), Add(Sqrt(w), Sqrt(z))))))),
    Variables(x, y, z, w),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(w, CC), GreaterEqual(Re(x), 0), GreaterEqual(Re(y), 0), GreaterEqual(Re(z), 0), Greater(Re(w), 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

Carlson symmetric elliptic integrals