Fungrim entry: 5e869b

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

Assumptions:

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C}

TeX:

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

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteEllipticE`	$E\!\left(\phi, m\right)$	Legendre incomplete elliptic integral of the second kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("5e869b"),
    Formula(Equal(IncompleteEllipticE(phi, m), Integral(Div(Sqrt(Sub(1, Mul(m, Pow(x, 2)))), Sqrt(Sub(1, Pow(x, 2)))), For(x, 0, Sin(phi))))),
    Variables(phi, m),
    Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, CC))))

Topics using this entry

Legendre elliptic integrals