# 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"),
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))))))