@stdlib/math-base-special-ellipe
v0.3.1TypeScript
Compute the complete elliptic integral of the second kind.
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Published by The Stdlib Authors
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ellipe
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> Compute the [complete elliptic integral of the second kind][elliptic-integral].
The [complete elliptic integral of the second kind][elliptic-integral] is defined as
where the parameter m is related to the modulus k by m = k^2.
Installation
``bash`
npm install @stdlib/math-base-special-ellipe
Usage
`javascript`
var ellipe = require( '@stdlib/math-base-special-ellipe' );
#### ellipe( m )
Computes the [complete elliptic integral of the second kind][elliptic-integral].
`javascript
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
`javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var ellipe = require( '@stdlib/math-base-special-ellipe' );
var opts = {
'dtype': 'float64'
};
var m = uniform( 100, -1.0, 1.0, opts );
logEachMap( 'ellipe(%0.4f) = %0.4f', m, ellipe );
`
*
C APIs
$3
`c`
#include "stdlib/math/base/special/ellipe.h"
#### stdlib_base_ellipe( m )
Computes the [complete elliptic integral of the second kind][elliptic-integral].
`c
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:
- m: [in] double input value.
`c`
double stdlib_base_ellipe( const double m );
$3
`c
#include "stdlib/math/base/special/ellipe.h"
#include
#include
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:2009a].
- 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][@fukushima:2015a].
*
See Also
- [@stdlib/math-base/special/ellipj][@stdlib/math/base/special/ellipj]: compute the Jacobi elliptic functions sn, cn, and dn.
- [@stdlib/math-base/special/ellipk][@stdlib/math/base/special/ellipk]: compute the complete elliptic integral of the first kind.
*
Notice
This package is part of [stdlib][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][stdlib], see the main project [repository][stdlib].
#### Community
[![Chat][chat-image]][chat-url]
---
Copyright
Copyright © 2016-2026. The Stdlib [Authors][stdlib-authors].
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[stdlib]: https://github.com/stdlib-js/stdlib
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[umd]: https://github.com/umdjs/umd
[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules
[deno-url]: https://github.com/stdlib-js/math-base-special-ellipe/tree/deno
[deno-readme]: https://github.com/stdlib-js/math-base-special-ellipe/blob/deno/README.md
[umd-url]: https://github.com/stdlib-js/math-base-special-ellipe/tree/umd
[umd-readme]: https://github.com/stdlib-js/math-base-special-ellipe/blob/umd/README.md
[esm-url]: https://github.com/stdlib-js/math-base-special-ellipe/tree/esm
[esm-readme]: https://github.com/stdlib-js/math-base-special-ellipe/blob/esm/README.md
[branches-url]: https://github.com/stdlib-js/math-base-special-ellipe/blob/main/branches.md
[elliptic-integral]: https://en.wikipedia.org/wiki/Elliptic_integral
[@fukushima:2009a]: https://doi.org/10.1007/s10569-009-9228-z
[@fukushima:2015a]: https://doi.org/10.1016/j.cam.2014.12.038
[@stdlib/math/base/special/ellipj]: https://www.npmjs.com/package/@stdlib/math-base-special-ellipj
[@stdlib/math/base/special/ellipk]: https://www.npmjs.com/package/@stdlib/math-base-special-ellipk