Fungrim entry: c10014

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

Assumptions:

n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right)

TeX:

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

n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right)

Definitions:

Fungrim symbol	Notation	Short description
`EllipticPi`	$\Pi\!\left(n, m\right)$	Legendre complete elliptic integral of the third kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("c10014"),
    Formula(Equal(EllipticPi(n, 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, 1)))),
    Variables(n, m),
    Assumptions(And(Element(n, OpenInterval(Neg(Infinity), 1)), Element(m, OpenInterval(Neg(Infinity), 1)))))

Topics using this entry

Legendre elliptic integrals