# 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))))

## Topics using this entry

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