Fungrim entry: 06223c

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteEllipticPi`	$\Pi\!\left(n, \phi, m\right)$	Legendre incomplete elliptic integral of the third kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`Sin`	$\sin(z)$	Sine
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`Pi`	$\pi$	The constant pi (3.14...)
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("06223c"),
    Formula(Equal(IncompleteEllipticPi(n, phi, m), Integral(Div(1, Mul(Mul(Sub(1, Mul(n, Pow(x, 2))), Sqrt(Sub(1, Pow(x, 2)))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, Sin(phi))))),
    Variables(n, phi, m),
    Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(n, OpenInterval(Neg(Infinity), 1)), Element(m, OpenInterval(Neg(Infinity), 1)))))

Topics using this entry

Legendre elliptic integrals