Fungrim entry: f0bcb5

E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right)

Assumptions:

m \in \mathbb{C} \setminus \left\{0, 1\right\}

TeX:

E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right)

m \in \mathbb{C} \setminus \left\{0, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteEllipticE`	$E\!\left(\phi, m\right)$	Legendre incomplete elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f0bcb5"),
    Formula(Equal(IncompleteEllipticE(Asin(Div(1, Sqrt(m))), m), Mul(Sqrt(m), Sub(EllipticE(Div(1, m)), Mul(Sub(1, Div(1, m)), EllipticK(Div(1, m))))))),
    Variables(m),
    Assumptions(Element(m, SetMinus(CC, Set(0, 1)))))

Topics using this entry

Legendre elliptic integrals