ellipticIntegralRF¶

Evaluates Carlson’s elliptic integral of the first kind \(R_F(x,y,z)\).

Synopsis¶

ellipticIntegralRF (x, y, z)

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 nonnegative.

Return Value¶

The complete elliptic integral \(R_F(x,y,z)\)

Description¶

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

\[R_F(x,y,z) = \tfrac{1}{2} \int_0^{\infty} \frac{dt}{[(t+x)(t+y)(t+z)]^{1/2}}\]

The arguments must be nonnegative and less than or equal to b/5. In addition, x + y, x + z, and y + z must be greater than or equal to 5s. Should any of these conditions fail, ellipticIntegralRF is set to b. Here, b = machine(2) is the largest and s = machine(1) is the smallest representable number. For more information, see the description for machine.

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

Example¶

The integral \(R_F(0,1,2)\) is computed.

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

x = 0.0
y = 1.0
z = 2.0
ans = ellipticIntegralRf(x, y, z)
print("RF(0, 1, 2) = %f" % (ans))

Output¶

RF(0, 1, 2) = 1.311029