Fungrim entry: 0455b3

K(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx

Assumptions:

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

TeX:

K(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, 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
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("0455b3"),
    Formula(Equal(EllipticK(m), Integral(Div(1, Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2))))), For(x, 0, Div(Pi, 2))))),
    Variables(m),
    Assumptions(Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity)))))

Topics using this entry

Legendre elliptic integrals