ellipk

Compute the complete elliptic integral of the first kind.

The complete elliptic integral of the first kind is defined as

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

Usage

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

ellipk( m )

Computes the complete elliptic integral of the first kind.

var v = ellipk( 0.5 );
// returns ~1.854

v = ellipk( -1.0 );
// returns ~1.311

v = ellipk( 2.0 );
// returns NaN

v = ellipk( Infinity );
// returns NaN

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

v = ellipk( NaN );
// returns NaN

Notes

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

Examples

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

var m;
var i;

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

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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.