# 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"),
Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), NotEqual(z, 0))))