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Fungrim entry: f48f54

E ⁣(ϕ,m)=sin(ϕ)RF ⁣(cos2 ⁣(ϕ),1msin2 ⁣(ϕ),1)13msin3 ⁣(ϕ)RD ⁣(cos2 ⁣(ϕ),1msin2 ⁣(ϕ),1)E\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) - \frac{1}{3} m \sin^{3}\!\left(\phi\right) R_D\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right)
Assumptions:ϕC  and  mC  and  π2Re(ϕ)π2\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2}
TeX:
E\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) - \frac{1}{3} m \sin^{3}\!\left(\phi\right) R_D\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right)

\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2}
Definitions:
Fungrim symbol Notation Short description
IncompleteEllipticEE ⁣(ϕ,m)E\!\left(\phi, m\right) Legendre incomplete elliptic integral of the second kind
Sinsin(z)\sin(z) Sine
CarlsonRFRF ⁣(x,y,z)R_F\!\left(x, y, z\right) Carlson symmetric elliptic integral of the first kind
Powab{a}^{b} Power
Coscos(z)\cos(z) Cosine
CarlsonRDRD ⁣(x,y,z)R_D\!\left(x, y, z\right) Degenerate Carlson symmetric elliptic integral of the third kind
CCC\mathbb{C} Complex numbers
Piπ\pi The constant pi (3.14...)
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("f48f54"),
    Formula(Equal(IncompleteEllipticE(phi, m), Sub(Mul(Sin(phi), CarlsonRF(Pow(Cos(phi), 2), Sub(1, Mul(m, Pow(Sin(phi), 2))), 1)), Mul(Mul(Mul(Div(1, 3), m), Pow(Sin(phi), 3)), CarlsonRD(Pow(Cos(phi), 2), Sub(1, Mul(m, Pow(Sin(phi), 2))), 1))))),
    Variables(phi, m),
    Assumptions(And(Element(phi, CC), Element(m, CC), LessEqual(Div(Neg(Pi), 2), Re(phi), Div(Pi, 2)))))

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2021-03-15 19:12:00.328586 UTC