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Fungrim entry: e54e61

RF ⁣(0,x,x)=1x(Γ ⁣(14))242π{1i,Im(x)<0  or  (Im(x)=0  and  Re(x)0)1+i,otherwiseR_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:xCx \in \mathbb{C}
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}
Fungrim symbol Notation Short description
CarlsonRFRF ⁣(x,y,z)R_F\!\left(x, y, z\right) Carlson symmetric elliptic integral of the first kind
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
GammaΓ(z)\Gamma(z) Gamma function
Piπ\pi The constant pi (3.14...)
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(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)))),
    Assumptions(Element(x, CC)))

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2021-03-15 19:12:00.328586 UTC