ellipe

Compute the complete elliptic integral of the second kind.

The complete elliptic integral of the second kind is defined as

where the parameter m is related to the modulus k by m = k^2.

Usage

var ellipe = require( '@stdlib/math/base/special/ellipe' );

ellipe( m )

Computes the complete elliptic integral of the second kind.

var v = ellipe( 0.5 );
// returns ~1.351

v = ellipe( -1.0 );
// returns ~1.910

v = ellipe( 2.0 );
// returns NaN

v = ellipe( Infinity );
// returns NaN

v = ellipe( -Infinity );
// returns NaN

v = ellipe( NaN );
// returns NaN

Notes

This function is valid for -∞ < m <= 1.

Examples

var randu = require( '@stdlib/random/base/randu' );
var ellipe = require( '@stdlib/math/base/special/ellipe' );

var m;
var i;

for ( i = 0; i < 100; i++ ) {
    m = -1.0 + ( randu() * 2.0 );
    console.log( 'ellipe(%d) = %d', m, ellipe( m ) );
}

References

Fukushima, Toshio. 2009. "Fast computation of complete elliptic integrals and Jacobian elliptic functions." Celestial Mechanics and Dynamical Astronomy 105 (4): 305. doi:10.1007/s10569-009-9228-z.
Fukushima, Toshio. 2015. "Precise and fast computation of complete elliptic integrals by piecewise minimax rational function approximation." Journal of Computational and Applied Mathematics 282 (July): 71–76. doi:10.1016/j.cam.2014.12.038.