# Fungrim entry: 02a8d7

$R_J\!\left(x, y, z, w\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + w\right) \sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, dt$
Assumptions:$x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)$
TeX:
R_J\!\left(x, y, z, w\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + w\right) \sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, 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) \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\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
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
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Source code for this entry:
Entry(ID("02a8d7"),
Formula(Equal(CarlsonRJ(x, y, z, w), Mul(Div(3, 2), Integral(Div(1, Mul(Mul(Mul(Add(t, w), Sqrt(Add(t, x))), Sqrt(Add(t, y))), Sqrt(Add(t, z)))), For(t, 0, Infinity))))),
Variables(x, y, z, w),
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))), Element(w, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC