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 ) );
}
C APIs
Usage
#include "stdlib/math/base/special/ellipe.h"
stdlib_base_ellipe( m )
Computes the complete elliptic integral of the second kind.
double out = stdlib_base_ellipe( 0.5 );
// returns ~1.351
out = stdlib_base_ellipe( -1.0 );
// returns ~1.910
The function accepts the following arguments:
- x:
[in] doubleinput value.
double stdlib_base_ellipe( const double m );
Examples
#include "stdlib/math/base/special/ellipe.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_ellipe( m );
printf( "ellipe(%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.