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)))))

Topics using this entry

Legendre elliptic integrals

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC