# Fungrim entry: dab889

$R_G\!\left(x, y, z\right) = \frac{1}{4} \int_{0}^{\infty} \frac{t}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) \, dt$
Assumptions:$x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right)$
TeX:
R_G\!\left(x, y, z\right) = \frac{1}{4} \int_{0}^{\infty} \frac{t}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) \, dt

x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonRG$R_G\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the second kind
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Sqrt$\sqrt{z}$ Principal square root
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
OpenInterval$\left(a, b\right)$ Open interval
Source code for this entry:
Entry(ID("dab889"),
Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenInterval(Neg(Infinity), 0))))))