# Fungrim entry: a203e9

$R_D\!\left(\lambda, x + \lambda, y + \lambda\right) + R_D\!\left(\mu, x + \mu, y + \mu\right) = R_D\!\left(0, x, y\right) - \frac{3}{y \sqrt{x + y + \lambda + \mu}}\; \text{ where } \mu = \frac{x y}{\lambda}$
Assumptions:$x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \lambda \in \mathbb{C} \setminus \left(-\infty, 0\right]$
TeX:
R_D\!\left(\lambda, x + \lambda, y + \lambda\right) + R_D\!\left(\mu, x + \mu, y + \mu\right) = R_D\!\left(0, x, y\right) - \frac{3}{y \sqrt{x + y + \lambda + \mu}}\; \text{ where } \mu = \frac{x y}{\lambda}

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \lambda \in \mathbb{C} \setminus \left(-\infty, 0\right]
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
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Source code for this entry:
Entry(ID("a203e9"),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)), Element(lamda, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))