# 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))))

## 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