# Fungrim entry: 649dc0

$R_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(y \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \notin \left(-\infty, 0\right)\right)\right)$
TeX:
R_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(y \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \notin \left(-\infty, 0\right)\right)\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonRF$R_F\!\left(x, y, z\right)$ 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("649dc0"),
Formula(Equal(CarlsonRF(x, x, y), Cases(Tuple(Div(Atan(Sqrt(Sub(Div(x, y), 1))), Sqrt(Sub(x, y))), NotEqual(x, y)), Tuple(Div(1, Sqrt(x)), Equal(x, y))))),
Variables(x, y),
Assumptions(And(Element(x, CC), Element(y, CC), Or(Element(y, OpenInterval(0, Infinity)), And(Element(x, OpenInterval(0, Infinity)), NotElement(y, OpenInterval(Neg(Infinity), 0)))))))

## 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