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"),
    Formula(Where(Equal(Add(CarlsonRD(lamda, Add(x, lamda), Add(y, lamda)), CarlsonRD(mu, Add(x, mu), Add(y, mu))), Sub(CarlsonRD(0, x, y), Div(3, Mul(y, Sqrt(Add(Add(Add(x, y), lamda), mu)))))), Def(mu, Div(Mul(x, y), lamda)))),
    Variables(x, y, lamda),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)), Element(lamda, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))

Topics using this entry

Carlson symmetric elliptic integrals