# 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)))))

## 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