Fungrim entry: bc2f88

R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)

Assumptions:

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)

TeX:

R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)

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
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("bc2f88"),
    Formula(Equal(CarlsonRC(Neg(x), y), Mul(Div(1, Sqrt(Add(x, y))), Sub(Div(Pi, 2), Mul(Atanh(Sqrt(Div(x, Add(x, y)))), ConstI))))),
    Variables(x, y),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

Topics using this entry

Specific values of Carlson symmetric elliptic integrals