Fungrim entry: 81fb10

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

Assumptions:

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right)

TeX:

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

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteEllipticF`	$F\!\left(\phi, m\right)$	Legendre incomplete 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
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`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("81fb10"),
    Formula(Equal(IncompleteEllipticF(phi, m), Integral(Div(1, Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2))))), For(x, 0, phi)))),
    Variables(phi, m),
    Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity))))))

Topics using this entry

Legendre elliptic integrals