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Fungrim entry: 5ada5f

RC ⁣(x,y)={atan ⁣(yx1)yx,x<y1x,x=yatanh ⁣(1yx)xy,x>yR_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(0,)  and  y(0,)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
CarlsonRCRC ⁣(x,y)R_C\!\left(x, y\right) Degenerate Carlson symmetric elliptic integral of the first kind
Atanatan(z)\operatorname{atan}(z) Inverse tangent
Sqrtz\sqrt{z} Principal square root
OpenInterval(a,b)\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)))))

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2021-03-15 19:12:00.328586 UTC