Fungrim entry: 5ada5f

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

Assumptions:

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

TeX:

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

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
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Sqrt`	$\sqrt{z}$	Principal square root
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("5ada5f"),
    Formula(Equal(CarlsonRC(x, y), Cases(Tuple(Div(Atan(Sqrt(Sub(Div(y, x), 1))), Sqrt(Sub(y, x))), Less(x, y)), Tuple(Div(1, Sqrt(x)), Equal(x, y)), Tuple(Div(Atanh(Sqrt(Sub(1, Div(y, x)))), Sqrt(Sub(x, y))), Greater(x, y))))),
    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