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Fungrim entry: 7c50d1

RG ⁣(0,x,x)=x2π3/22(Γ ⁣(14))2{1+i,Im(x)<0  or  (Im(x)=0  and  Re(x)0)1i,otherwiseR_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:xCx \in \mathbb{C}
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}
Fungrim symbol Notation Short description
CarlsonRGRG ⁣(x,y,z)R_G\!\left(x, y, z\right) Carlson symmetric elliptic integral of the second kind
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
GammaΓ(z)\Gamma(z) Gamma function
ConstIii Imaginary unit
ImIm(z)\operatorname{Im}(z) Imaginary part
ReRe(z)\operatorname{Re}(z) Real part
CCC\mathbb{C} Complex numbers
Source code for this entry:
    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))))),
    Assumptions(Element(x, CC)))

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2021-03-15 19:12:00.328586 UTC