Fungrim entry: 7c50d1

R_G\!\left(0, x, -x\right) = \sqrt{x} \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \begin{cases} 1 + i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 - i, & \text{otherwise}\\ \end{cases}

Assumptions:

x \in \mathbb{C}

TeX:

R_G\!\left(0, x, -x\right) = \sqrt{x} \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \begin{cases} 1 + i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 - i, & \text{otherwise}\\ \end{cases}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7c50d1"),
    Formula(Equal(CarlsonRG(0, x, Neg(x)), Mul(Mul(Sqrt(x), Div(Mul(Sqrt(2), Pow(Pi, Div(3, 2))), Mul(2, Pow(Gamma(Div(1, 4)), 2)))), Cases(Tuple(Add(1, ConstI), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Sub(1, ConstI), Otherwise))))),
    Variables(x),
    Assumptions(Element(x, CC)))

Topics using this entry

Specific values of Carlson symmetric elliptic integrals