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"),
    Formula(Where(Equal(Add(CarlsonRJ(Add(x, lamda), Add(y, lamda), lamda, Add(w, lamda)), CarlsonRJ(Add(x, mu), Add(y, mu), mu, Add(w, mu))), Sub(CarlsonRJ(x, y, 0, w), Mul(3, CarlsonRC(Mul(Pow(w, 2), Add(Add(Add(lamda, mu), x), y)), Mul(Mul(w, Add(w, lamda)), Add(w, mu)))))), Def(mu, Div(Mul(x, y), lamda)))),
    Variables(x, y, w, lamda),
    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))))))

Topics using this entry

Carlson symmetric elliptic integrals