Fungrim entry: 47dead

K(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx

Assumptions:

m \in \mathbb{C} \setminus \left[1, \infty\right)

TeX:

K(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx

m \in \mathbb{C} \setminus \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("47dead"),
    Formula(Equal(EllipticK(m), Integral(Div(1, Mul(Sqrt(Sub(1, Pow(x, 2))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, 1)))),
    Variables(m),
    Assumptions(Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity)))))

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