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ellipk

Compute the complete elliptic integral of the first kind.

The complete elliptic integral of the first kind is defined as

$$K(m)=\int_0^\tfrac{\pi}{2} \frac{d\theta}{\sqrt{1-m \sin^2\theta}}$$

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 uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var ellipk = require( '@stdlib/math/base/special/ellipk' );

var opts = {
    'dtype': 'float64'
};
var m = uniform( 100, -1.0, 1.0, opts );

logEachMap( 'ellipk(%0.4f) = %0.4f', m, ellipk );

C APIs

Usage

#include "stdlib/math/base/special/ellipk.h"

stdlib_base_ellipk( m )

Computes the complete elliptic integral of the first kind.

double out = stdlib_base_ellipk( 0.5 );
// returns ~1.854

out = stdlib_base_ellipk( -1.0 );
// returns ~1.311

The function accepts the following arguments:

  • x: [in] double input value.
double stdlib_base_ellipk( const double m );

Examples

#include "stdlib/math/base/special/ellipk.h"
#include <stdlib.h>
#include <stdio.h>

int main( void ) {
    double m;
    double v;
    int i;

    for ( i = 0; i < 100; i++ ) {
        m = -1.0 + ( ( (double)rand() / (double)RAND_MAX ) * 2.0 );
        v = stdlib_base_ellipk( m );
        printf( "ellipk(%lf) = %lf\n", m, v );
    }
}

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.

See Also