Fungrim entry: f48f54

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:

\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
`IncompleteEllipticE`	$E\!\left(\phi, m\right)$	Legendre incomplete elliptic integral of the second kind
`Sin`	$\sin(z)$	Sine
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`CarlsonRD`	$R_D\!\left(x, y, z\right)$	Degenerate Carlson symmetric elliptic integral of the third kind
`CC`	$\mathbb{C}$	Complex numbers
`Pi`	$\pi$	The constant pi (3.14...)
`Re`	$\operatorname{Re}(z)$	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)))))

Fungrim entry: f48f54

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