Fungrim entry: 9b0385

\Pi\!\left(\frac{1}{2}, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} + \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

\Pi\!\left(\frac{1}{2}, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} + \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`EllipticPi`	$\Pi\!\left(n, m\right)$	Legendre complete elliptic integral of the third kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("9b0385"),
    Formula(Equal(EllipticPi(Div(1, 2), Div(1, 2)), Add(Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Pi))), Div(Mul(2, Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2))))))

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