ellipticIntegralRJ¶

Evaluates Carlson’s elliptic integral of the third kind \(R_J(x,y,z, \rho)\).

Synopsis¶

ellipticIntegralRJ (x, y, z, rho)

Required Arguments¶

float x (Input): First variable of the incomplete elliptic integral. It must be nonnegative.
float y (Input): Second variable of the incomplete elliptic integral. It must be nonnegative.
float z (Input): Third variable of the incomplete elliptic integral. It must be positive.
float rho (Input): Fourth variable of the incomplete elliptic integral. It must be positive.

Return Value¶

The complete elliptic integral \(R_J(x,y,z,\rho)\).

Description¶

Carlson’s elliptic integral of the third kind is defined to be

\[R_J(x,y,z,\rho) = \tfrac{3}{2} \int_0^{\infty} \frac{dt}{\left[(t+x)(t+y)(t+z)(t+\rho)^2\right]^{1/2}}\]

The arguments must be nonnegative. In addition, \(x+y\), \(x+z\), \(y+z\) and ρ must be greater than or equal to \((5s)^{1/3}\) and less than or equal to \(0.3(b/5)^{1/3}\), where s = machine(1) is the smallest representable floating-point number. Should any of these conditions fail, ellipticIntegralRJ is set to b = machine(2), the largest floating-point number. For more information, see the description for machine.

The function ellipticIntegralRJ is based on the code by Carlson and Notis (1981) and the work of Carlson (1979).

Example¶

The integral \(R_J(2,3,4,5)\) is computed.

from __future__ import print_function
from numpy import *
from pyimsl.math.ellipticIntegralRj import ellipticIntegralRj

x = 2.0
y = 3.0
z = 4.0
rho = 5.0

ans = ellipticIntegralRj(x, y, z, rho)
print("RJ(2, 3, 4, 5) = %f" % ans)

Output¶

RJ(2, 3, 4, 5) = 0.142976