ellipticIntegralRC

Evaluates an elementary integral from which inverse circular functions, logarithms and inverse hyperbolic functions can be computed.

Synopsis

ellipticIntegralRC (x,  y)

Required Arguments

float x (Input)

First variable of the incomplete elliptic integral. It must be nonnegative and must satisfy the conditions given below.

float y (Input)

Second variable of the incomplete elliptic integral. It must be positive and must satisfy the conditions given below.

Return Value

The elliptic integral $R_C(x,y)$.

Description

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

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

The argument x must be nonnegative, y must be positive, and x +y must be less than or equal to b/5 and greater than or equal to 5s. If any of these conditions are false, the ellipticIntegralRC is set to b. Here, b = machine(2) is the largest and s = ­machine(1) is the smallest representable floating-point number. For more information, see the description for machine.

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

Example

The integral $R_C(2.25,2)$ is computed.

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

x = 2.25
y = 2.0
ans = ellipticIntegralRc(x, y)
print("RC(2.25, 2.0) = %f" % (ans))

Output

RC(2.25, 2.0) = 0.693147