Fungrim entry: 8f4e31

\Pi\!\left(n, \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} n \sin^{3}\!\left(\phi\right) R_J\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1, 1 - n \sin^{2}\!\left(\phi\right)\right)

Assumptions:

n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \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:

\Pi\!\left(n, \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} n \sin^{3}\!\left(\phi\right) R_J\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1, 1 - n \sin^{2}\!\left(\phi\right)\right)

n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \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
`IncompleteEllipticPi`	$\Pi\!\left(n, \phi, m\right)$	Legendre incomplete elliptic integral of the third 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
`CarlsonRJ`	$R_J\!\left(x, y, z, w\right)$	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("8f4e31"),
    Formula(Equal(IncompleteEllipticPi(n, phi, m), Add(Mul(Sin(phi), CarlsonRF(Pow(Cos(phi), 2), Sub(1, Mul(m, Pow(Sin(phi), 2))), 1)), Mul(Mul(Mul(Div(1, 3), n), Pow(Sin(phi), 3)), CarlsonRJ(Pow(Cos(phi), 2), Sub(1, Mul(m, Pow(Sin(phi), 2))), 1, Sub(1, Mul(n, Pow(Sin(phi), 2)))))))),
    Variables(n, phi, m),
    Assumptions(And(Element(n, CC), Element(phi, CC), Element(m, CC), LessEqual(Div(Neg(Pi), 2), Re(phi), Div(Pi, 2)))))

Fungrim entry: 8f4e31

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