# Fungrim entry: 8c9ba1

$R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \end{cases}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left(0, \infty\right)$
TeX:
R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \end{cases}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \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
Atan$\operatorname{atan}(z)$ Inverse tangent
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("8c9ba1"),
Formula(Equal(CarlsonRC(x, Mul(c, x)), Cases(Tuple(Div(Atan(Sqrt(Sub(c, 1))), Sqrt(Mul(Sub(c, 1), x))), Greater(c, 1)), Tuple(Div(1, Sqrt(x)), Equal(c, 1)), Tuple(Div(Atanh(Sqrt(Sub(1, c))), Sqrt(Mul(Sub(1, c), x))), Less(c, 1))))),
Variables(x, c),
Assumptions(And(Element(x, CC), Element(c, 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