Fungrim entry: 8c9ba1

R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \end{cases}

Assumptions:

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

TeX:

R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \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
`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("8c9ba1"),
    Formula(Equal(CarlsonRC(x, Mul(c, x)), Cases(Tuple(Div(Atan(Sqrt(Sub(c, 1))), Sqrt(Mul(Sub(c, 1), x))), Greater(c, 1)), Tuple(Div(1, Sqrt(x)), Equal(c, 1)), Tuple(Div(Atanh(Sqrt(Sub(1, c))), Sqrt(Mul(Sub(1, c), x))), Less(c, 1))))),
    Variables(x, c),
    Assumptions(And(Element(x, CC), Element(c, OpenInterval(0, Infinity)))))

Topics using this entry

Specific values of Carlson symmetric elliptic integrals