`ellipticPi`

Complete and incomplete elliptic integrals of the third kind.

Category: Mathematics > Mathematical Functions

Syntax

P = ellipticPi(n, m)

Complete integral of the third kind.

P = ellipticPi(n, phi, m)

Incomplete integral with amplitude phi in radians.

Description

The integral is integral from 0 to phi of 1/((1 - n*sin(t)^2)*sqrt(1 - m*sin(t)^2)) dt, with phi = pi/2 for the complete form. For n = 0 it reduces to ellipticF(phi, m), and for n = m it can be written with ellipticE. It appears in the motion of a body in a central field and in magnetostatics.

Input arguments

n: Characteristic.
phi: Amplitude angle in radians.
m: Parameter (m = k^2).

Output arguments

P: Array formed from the broadcast inputs.

Example

Complete and incomplete forms

ellipticPi(0.2, 0.5)
ellipticPi(0.2, pi/4, 0.5)
ellipticPi(0, 0.5) - ellipticK(0.5)

Output

ans = 2.0930
ans = 0.8593
ans = 0

Notes

Some complex-valued cases, such as the complete integral with n > 1, return NaN in MatLivre.

ellipticPi