ellipe

Compute the complete elliptic integral of the second kind.

The complete elliptic integral of the second kind is defined as

E(m)=\int_0^{\pi/2} \sqrt{1 - m (\sin\theta)^2} d\theta

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

Installation

npm install @stdlib/math-base-special-ellipe

Alternatively,

To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
If you are using Deno, visit the deno branch (see README for usage intructions).
For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

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] double input 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.

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.