https://github.com/stdlib-js/math-base-special-ellipj

Compute the Jacobi elliptic functions sn, cn, and dn.
https://github.com/stdlib-js/math-base-special-ellipj

javascript node node-js nodejs stdlib

Last synced: 4 months ago
JSON representation

Compute the Jacobi elliptic functions sn, cn, and dn.

Host: GitHub
URL: https://github.com/stdlib-js/math-base-special-ellipj
Owner: stdlib-js
License: apache-2.0
Created: 2022-12-18T01:59:03.000Z (almost 3 years ago)
Default Branch: main
Last Pushed: 2025-06-12T01:46:09.000Z (5 months ago)
Last Synced: 2025-06-12T02:48:55.614Z (5 months ago)
Topics: javascript, node, node-js, nodejs, stdlib
Language: JavaScript
Homepage: https://github.com/stdlib-js/stdlib
Size: 1.71 MB
Stars: 4
Watchers: 3
Forks: 0
Open Issues: 0
Metadata Files:
- Readme: README.md
- Changelog: CHANGELOG.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
- Code of conduct: CODE_OF_CONDUCT.md
- Citation: CITATION.cff
- Security: SECURITY.md

Awesome Lists containing this project

README

          

  

    About stdlib...

  

  
We believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.

  The library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.

  When you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.

  To join us in bringing numerical computing to the web, get started by checking us out on GitHub, and please consider financially supporting stdlib. We greatly appreciate your continued support!


# ellipj

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] 

> Compute the [Jacobi elliptic functions][jacobi-elliptic] sn, cn, and dn.

The [Jacobi elliptic functions][jacobi-elliptic] may be defined as the inverse of the [incomplete elliptic integral of the first kind][incomplete-elliptic]. Accordingly, they compute the value `φ` which satisfies the equation

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

## Installation

```bash

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

```

Alternatively,

-   To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).

-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).

-   For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).

The [branches.md][branches-url] 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

```javascript

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

```

#### ellipj( u, m )

Computes the [Jacobi elliptic functions][jacobi-elliptic] functions `sn`, `cn`, and `dn`, and the Jacobi amplitude `am`.

```javascript

var v = ellipj( 0.3, 0.5 );

// returns [ ~0.293, ~0.956, ~0.978, ~0.298 ]

v = ellipj( 0.0, 0.0 );

// returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]

v = ellipj( Infinity, 1.0 );

// returns [ ~1.0, ~0.0, ~0.0, ~1.571 ]

v = ellipj( 0.0, -2.0 );

// returns [ ~0.0, ~1.0, ~1.0, NaN ]

v = ellipj( NaN, NaN );

// returns [ NaN, NaN, NaN, NaN ]

```

#### ellipj.assign( u, m, out, stride, offset )

Computes the Jacobi elliptic functions `sn`, `cn`, `dn`, and Jacobi amplitude `am` and assigns results to a provided output array.

```javascript

var Float64Array = require( '@stdlib/array-float64' );

var out = new Float64Array( 4 );

var v = ellipj.assign( 0.0, 0.0, out, 1, 0 );

// returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]

var bool = ( v === out );

// returns true

```

#### ellipj.sn( u, m )

Computes the Jacobi elliptic function `sn` of value `u` with modulus `m`.

```javascript

var v = ellipj.sn( 0.3, 0.5 );

// returns ~0.293

```

#### ellipj.cn( u, m )

Computes the Jacobi elliptic function `cn` of value `u` with modulus `m`.

```javascript

var v = ellipj.cn( 0.3, 0.5 );

// returns ~0.956

```

#### ellipj.dn( u, m )

Computes the Jacobi elliptic function `dn` of value `u` with modulus `m`.

```javascript

var v = ellipj.dn( 0.3, 0.5 );

// returns ~0.978

```

#### ellipj.am( u, m )

Computes the Jacobi amplitude `am` of value `u` with modulus `m`.

```javascript

var v = ellipj.am( 0.3, 0.5 );

// returns ~0.298

v = ellipj.am( 0.3, 2.0 );

// returns NaN

```

Although `sn`, `cn`, and `dn` may be computed for `-∞ < m < ∞`, the domain of `am` is `0 ≤ m ≤ 1`. For `m < 0` or `m > 1`, the function returns `NaN`.

## Notes

-   Functions `sn`, `cn`, and `dn` are valid for `-∞ < m < ∞`. Values for `m < 0` or `m > 1` are computed in terms of Jacobi elliptic functions with `0 < m < 1` via the transformations outlined in Equations 16.13 and 16.15 from _The Handbook of Mathematical Functions_ (Abramowitz and Stegun).

-   If more than one of `sn`, `cn`, `dn`, or `am` is to be computed, preferring using `ellipj` to compute all four values simultaneously.

## Examples

```javascript

var linspace = require( '@stdlib/array-base-linspace' );

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

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

var m = 0.7;

var u = linspace( 0.0, ellipk( m ), 100 );

var out;

var i;

for ( i = 0; i < 100; i++ ) {

    out = ellipj( u[ i ], m );

    console.log( 'sn(%d, %d) = %d', u[ i ], m, out[ 0 ] );

    console.log( 'cn(%d, %d) = %d', u[ i ], m, out[ 1 ] );

    console.log( 'dn(%d, %d) = %d', u[ i ], m, out[ 2 ] );

    console.log( 'am(%d, %d) = %d', u[ i ], m, out[ 3 ] );

}

```

* * *

## C APIs

### Usage

```c

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

```

#### stdlib_base_ellipj( x, m, &sn, &cn, &dn, &am )

Computes the [Jacobi elliptic functions][jacobi-elliptic] functions `sn`, `cn`, and `dn`, and the Jacobi amplitude `am`.

```c

double sn;

double cn;

double dn;

double am;

stdlib_base_ellipj( 0.3, 0.5, &sn, &cn, &dn, &am );

```

The function accepts the following arguments:

-   **x**: `[in] double` input value.

-   **m**: `[in] double` modulus `m`, equivalent to `k²`.

-   **sn**: `[out] double*` destination for the sine amplitude.

-   **cn**: `[out] double*` destination for the cosine amplitude.

-   **dn**: `[out] double*` destination for the delta amplitude.

-   **am**: `[out] double*` destination for the Jacobi amplitude.

```c

void stdlib_base_ellipj( const double u, const double m, double* sn, double* cn, double* dn, double* am );

```

### Examples

```c

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

#include 

#include 

int main( void ) {

    double sn;

    double cn;

    double dn;

    double am;

    double x;

    int i;

    for ( i = 0; i < 100; i++ ) {

        x = 2.0 * ( (double)rand() / (double)RAND_MAX );

        stdlib_base_ellipj( x, 0.7, &sn, &cn, &dn, &am );

        printf( "x: %lf, m: %lf => sn: %lf, cn: %lf, dn: %lf, am: %lf\n", x, 0.7, sn, cn, dn, am );

    }

}

```

* * *

## 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/ellipe`][@stdlib/math/base/special/ellipe]: compute the complete elliptic integral of the second kind.

-   [`@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]

---

## License

See [LICENSE][stdlib-license].

## Copyright

Copyright © 2016-2025. The Stdlib [Authors][stdlib-authors].

[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-ellipj.svg

[npm-url]: https://npmjs.org/package/@stdlib/math-base-special-ellipj

[test-image]: https://github.com/stdlib-js/math-base-special-ellipj/actions/workflows/test.yml/badge.svg?branch=main

[test-url]: https://github.com/stdlib-js/math-base-special-ellipj/actions/workflows/test.yml?query=branch:main

[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-ellipj/main.svg

[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-ellipj?branch=main

[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg

[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im

[stdlib]: https://github.com/stdlib-js/stdlib

[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors

[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-ellipj/tree/deno

[deno-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/deno/README.md

[umd-url]: https://github.com/stdlib-js/math-base-special-ellipj/tree/umd

[umd-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/umd/README.md

[esm-url]: https://github.com/stdlib-js/math-base-special-ellipj/tree/esm

[esm-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/esm/README.md

[branches-url]: https://github.com/stdlib-js/math-base-special-ellipj/blob/main/branches.md

[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-ellipj/main/LICENSE

[jacobi-elliptic]: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions

[incomplete-elliptic]: 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/ellipe]: https://github.com/stdlib-js/math-base-special-ellipe

[@stdlib/math/base/special/ellipk]: https://github.com/stdlib-js/math-base-special-ellipk

ecosyste.ms

Data

Tools

Indexes

Applications

Experiments

Awesome

https://github.com/stdlib-js/math-base-special-ellipj

Awesome Lists containing this project

README