Fungrim entry: d4b12e

R_J\!\left(x, y, y, w\right) = \begin{cases} \frac{3}{w - y} \left(R_C\!\left(x, y\right) - R_C\!\left(x, w\right)\right), & y \ne w\\\frac{3}{2 \left(y - x\right)} \left(R_C\!\left(x, y\right) - \frac{\sqrt{x}}{y}\right), & y = w \;\mathbin{\operatorname{and}}\; x \ne y\\{x}^{-3 / 2}, & x = y = w\\ \end{cases}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C}

TeX:

R_J\!\left(x, y, y, w\right) = \begin{cases} \frac{3}{w - y} \left(R_C\!\left(x, y\right) - R_C\!\left(x, w\right)\right), & y \ne w\\\frac{3}{2 \left(y - x\right)} \left(R_C\!\left(x, y\right) - \frac{\sqrt{x}}{y}\right), & y = w \;\mathbin{\operatorname{and}}\; x \ne y\\{x}^{-3 / 2}, & x = y = w\\ \end{cases}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C}

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
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d4b12e"),
    Formula(Equal(CarlsonRJ(x, y, y, w), Cases(Tuple(Mul(Div(3, Sub(w, y)), Sub(CarlsonRC(x, y), CarlsonRC(x, w))), NotEqual(y, w)), Tuple(Mul(Div(3, Mul(2, Sub(y, x))), Sub(CarlsonRC(x, y), Div(Sqrt(x), y))), And(Equal(y, w), NotEqual(x, y))), Tuple(Pow(x, Neg(Div(3, 2))), Equal(x, y, w))))),
    Variables(x, y, w),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(w, CC))))

Fungrim entry: d4b12e

Topics using this entry