# Fungrim entry: 4eac3f

$R_J\!\left(x + \lambda, y + \lambda, \lambda, w + \lambda\right) + R_J\!\left(x + \mu, y + \mu, \mu, w + \mu\right) = R_J\!\left(x, y, 0, w\right) - 3 R_C\!\left({w}^{2} \left(\lambda + \mu + x + y\right), w \left(w + \lambda\right) \left(w + \mu\right)\right)\; \text{ where } \mu = \frac{x y}{\lambda}$
Assumptions:$x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; w \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \lambda \in \mathbb{C} \setminus \left(-\infty, 0\right]$
TeX:
R_J\!\left(x + \lambda, y + \lambda, \lambda, w + \lambda\right) + R_J\!\left(x + \mu, y + \mu, \mu, w + \mu\right) = R_J\!\left(x, y, 0, w\right) - 3 R_C\!\left({w}^{2} \left(\lambda + \mu + x + y\right), w \left(w + \lambda\right) \left(w + \mu\right)\right)\; \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}}\; w \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \lambda \in \mathbb{C} \setminus \left(-\infty, 0\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
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("4eac3f"),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)), Element(w, OpenInterval(0, Infinity)), Element(lamda, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))