Fungrim entry: e54e61

R_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \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_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \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
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)
`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("e54e61"),
    Formula(Mul(Mul(Equal(CarlsonRF(0, x, Neg(x)), Div(1, Sqrt(x))), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi))))), Cases(Tuple(Sub(1, ConstI), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(1, ConstI), Otherwise)))),
    Variables(x),
    Assumptions(Element(x, CC)))

Topics using this entry

Specific values of Carlson symmetric elliptic integrals