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Fungrim entry: 60f858

Π ⁣(n,ϕ,m)=0ϕ1(1nsin2 ⁣(x))1msin2 ⁣(x)dx\Pi\!\left(n, \phi, m\right) = \int_{0}^{\phi} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
Assumptions:ϕ[π2,π2]  and  n(,1)  and  m(,1)\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)
\Pi\!\left(n, \phi, m\right) = \int_{0}^{\phi} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, 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)
Fungrim symbol Notation Short description
IncompleteEllipticPiΠ ⁣(n,ϕ,m)\Pi\!\left(n, \phi, m\right) Legendre incomplete elliptic integral of the third kind
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Sinsin(z)\sin(z) Sine
Sqrtz\sqrt{z} Principal square root
ClosedInterval[a,b]\left[a, b\right] Closed interval
Piπ\pi The constant pi (3.14...)
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Equal(IncompleteEllipticPi(n, phi, m), Integral(Div(1, Mul(Sub(1, Mul(n, Pow(Sin(x), 2))), Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2)))))), For(x, 0, 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)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC