Fungrim entry: 9ccaef

\Pi\!\left(n, m\right) = R_F\!\left(0, 1 - m, 1\right) + \frac{n}{3} R_J\!\left(0, 1 - m, 1, 1 - n\right)

Assumptions:

n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \ne 1

TeX:

\Pi\!\left(n, m\right) = R_F\!\left(0, 1 - m, 1\right) + \frac{n}{3} R_J\!\left(0, 1 - m, 1, 1 - n\right)

n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \ne 1

Definitions:

Fungrim symbol	Notation	Short description
`EllipticPi`	$\Pi\!\left(n, m\right)$	Legendre complete elliptic integral of the third kind
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`CarlsonRJ`	$R_J\!\left(x, y, z, w\right)$	Carlson symmetric elliptic integral of the third kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9ccaef"),
    Formula(Equal(EllipticPi(n, m), Add(CarlsonRF(0, Sub(1, m), 1), Mul(Div(n, 3), CarlsonRJ(0, Sub(1, m), 1, Sub(1, n)))))),
    Variables(n, m),
    Assumptions(And(Element(n, CC), Element(m, CC), NotEqual(m, 1))))

Fungrim entry: 9ccaef

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