# Fungrim entry: bc2f88

$R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)$
Assumptions:$x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)$
TeX:
R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonRC$R_C\!\left(x, y\right)$ Degenerate Carlson symmetric elliptic integral of the first kind
Sqrt$\sqrt{z}$ Principal square root
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("bc2f88"),
Formula(Equal(CarlsonRC(Neg(x), y), Mul(Div(1, Sqrt(Add(x, y))), Sub(Div(Pi, 2), Mul(Atanh(Sqrt(Div(x, Add(x, y)))), ConstI))))),
Variables(x, y),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

## 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