Fungrim entry: 649dc0

R_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(y \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \notin \left(-\infty, 0\right)\right)\right)

TeX:

R_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(y \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \notin \left(-\infty, 0\right)\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	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("649dc0"),
    Formula(Equal(CarlsonRF(x, x, y), Cases(Tuple(Div(Atan(Sqrt(Sub(Div(x, y), 1))), Sqrt(Sub(x, y))), NotEqual(x, y)), Tuple(Div(1, Sqrt(x)), Equal(x, y))))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC), Or(Element(y, OpenInterval(0, Infinity)), And(Element(x, OpenInterval(0, Infinity)), NotElement(y, OpenInterval(Neg(Infinity), 0)))))))

Fungrim entry: 649dc0

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