ellipticIntegralE

Evaluates the complete elliptic integral of the second kind E(x).

Synopsis

ellipticIntegralE (x)

Required Arguments

float x (Input)

Argument for which the function value is desired.

Return Value

The complete elliptic integral E(x).

Description

The complete elliptic integral of the second kind is defined to be

$$E(x) = \int_0^{\pi/2} \left[1 - x \sin^2 \theta\right]^{1/2} d \theta \text{ for } 0 \leq x < 1$$

The argument x must satisfy 0 ≤ x < 1; otherwise, ellipticIntegralE returns machine(2), the largest representable floating-point number. For more information, see the description for machine.

The function E(x) is computed using the routine ellipticIntegralRF and ellipticIntegralRD. The computation is done using the relation

$$E(x) = R_F(0,1-x,1) - \tfrac{x}{3} R_D(0,1-x,1)$$

Example

The integral E(0.33) is evaluated.

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

x = 0.33
x = ellipticIntegralE(x)
print("E(0.33) = %f" % (x))

Output

E(0.33) = 1.431832