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Fungrim entry: 33ee4a

F ⁣(ϕ,m)=0sin(ϕ)11x21mx2dxF\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
Assumptions:ϕ[π2,π2]  and  mC[1,)\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right)
F\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx

\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right)
Fungrim symbol Notation Short description
IncompleteEllipticFF ⁣(ϕ,m)F\!\left(\phi, m\right) Legendre incomplete elliptic integral of the first kind
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
Sinsin(z)\sin(z) Sine
ClosedInterval[a,b]\left[a, b\right] Closed interval
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Equal(IncompleteEllipticF(phi, m), Integral(Div(1, Mul(Sqrt(Sub(1, Pow(x, 2))), Sqrt(Sub(1, Mul(m, 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, SetMinus(CC, ClosedOpenInterval(1, Infinity))))))

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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