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# Fungrim entry: 175b7a

$K\!\left(\frac{1 - \sqrt{3} i}{2}\right) = \frac{{e}^{-i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi}$
TeX:
K\!\left(\frac{1 - \sqrt{3} i}{2}\right) = \frac{{e}^{-i \pi / 12} \cdot  {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi}
Definitions:
Fungrim symbol Notation Short description
EllipticK$K(m)$ Legendre complete elliptic integral of the first kind
Sqrt$\sqrt{z}$ Principal square root
ConstI$i$ Imaginary unit
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
Gamma$\Gamma(z)$ Gamma function
Source code for this entry:
Entry(ID("175b7a"),
Formula(Equal(EllipticK(Div(Sub(1, Mul(Sqrt(3), ConstI)), 2)), Div(Mul(Mul(Exp(Neg(Div(Mul(ConstI, Pi), 12))), Pow(3, Div(1, 4))), Pow(Gamma(Div(1, 3)), 3)), Mul(Pow(2, Div(7, 3)), Pi)))))

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2021-03-15 19:12:00.328586 UTC