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Fungrim entry: 175b7a

K ⁣(13i2)=eiπ/1231/4(Γ ⁣(13))327/3πK\!\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
EllipticKK(m)K(m) Legendre complete elliptic integral of the first kind
Sqrtz\sqrt{z} Principal square root
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
GammaΓ(z)\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