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

Specific values of Carlson symmetric elliptic integrals