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

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
Atan$\operatorname{atan}(z)$ Inverse tangent
Sqrt$\sqrt{z}$ Principal square root
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("5ada5f"),
Formula(Equal(CarlsonRC(x, y), Cases(Tuple(Div(Atan(Sqrt(Sub(Div(y, x), 1))), Sqrt(Sub(y, x))), Less(x, y)), Tuple(Div(1, Sqrt(x)), Equal(x, y)), Tuple(Div(Atanh(Sqrt(Sub(1, Div(y, x)))), Sqrt(Sub(x, y))), Greater(x, y))))),
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