# Fungrim entry: 7348e3

$R_C\!\left(x, -c x\right) = \frac{1}{\sqrt{\left(c + 1\right) x}} \begin{cases} \operatorname{atanh}\!\left(\sqrt{c + 1}\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\\operatorname{atanh}\!\left(\sqrt{c + 1}\right) + \pi i, & \text{otherwise}\\ \end{cases}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left(0, \infty\right)$
TeX:
R_C\!\left(x, -c x\right) = \frac{1}{\sqrt{\left(c + 1\right) x}} \begin{cases} \operatorname{atanh}\!\left(\sqrt{c + 1}\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\\operatorname{atanh}\!\left(\sqrt{c + 1}\right) + \pi i, & \text{otherwise}\\ \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
Sqrt$\sqrt{z}$ Principal square root
Im$\operatorname{Im}(z)$ Imaginary part
Re$\operatorname{Re}(z)$ Real part
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("7348e3"),
Formula(Equal(CarlsonRC(x, Neg(Mul(c, x))), Mul(Div(1, Sqrt(Mul(Add(c, 1), x))), Cases(Tuple(Atanh(Sqrt(Add(c, 1))), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(Atanh(Sqrt(Add(c, 1))), Mul(Pi, ConstI)), Otherwise))))),
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