{"id":26119141,"url":"https://github.com/stdlib-js/math-base-special-ellipj","last_synced_at":"2025-07-13T16:32:44.021Z","repository":{"id":65696075,"uuid":"579532812","full_name":"stdlib-js/math-base-special-ellipj","owner":"stdlib-js","description":"Compute the Jacobi elliptic functions sn, cn, and dn.","archived":false,"fork":false,"pushed_at":"2025-06-12T01:46:09.000Z","size":1794,"stargazers_count":4,"open_issues_count":0,"forks_count":0,"subscribers_count":3,"default_branch":"main","last_synced_at":"2025-06-12T02:48:55.614Z","etag":null,"topics":["javascript","node","node-js","nodejs","stdlib"],"latest_commit_sha":null,"homepage":"https://github.com/stdlib-js/stdlib","language":"JavaScript","has_issues":false,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"apache-2.0","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/stdlib-js.png","metadata":{"files":{"readme":"README.md","changelog":"CHANGELOG.md","contributing":"CONTRIBUTING.md","funding":null,"license":"LICENSE","code_of_conduct":"CODE_OF_CONDUCT.md","threat_model":null,"audit":null,"citation":"CITATION.cff","codeowners":null,"security":"SECURITY.md","support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null},"funding":{"github":["stdlib-js"],"open_collective":"stdlib","tidelift":"npm/@stdlib/stdlib"}},"created_at":"2022-12-18T01:59:03.000Z","updated_at":"2025-06-12T01:44:03.000Z","dependencies_parsed_at":"2023-10-05T10:14:51.001Z","dependency_job_id":"b3f371ca-fc7f-46a7-8376-8bef9e5ceb5a","html_url":"https://github.com/stdlib-js/math-base-special-ellipj","commit_stats":{"total_commits":14,"total_committers":1,"mean_commits":14.0,"dds":0.0,"last_synced_commit":"a1788dcb5a154a88e73c2e44b13f5512f2db75f3"},"previous_names":[],"tags_count":24,"template":false,"template_full_name":null,"purl":"pkg:github/stdlib-js/math-base-special-ellipj","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fmath-base-special-ellipj","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fmath-base-special-ellipj/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fmath-base-special-ellipj/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fmath-base-special-ellipj/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/stdlib-js","download_url":"https://codeload.github.com/stdlib-js/math-base-special-ellipj/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fmath-base-special-ellipj/sbom","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":265173403,"owners_count":23722560,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["javascript","node","node-js","nodejs","stdlib"],"created_at":"2025-03-10T12:15:43.220Z","updated_at":"2025-07-13T16:32:44.009Z","avatar_url":"https://github.com/stdlib-js.png","language":"JavaScript","funding_links":["https://github.com/sponsors/stdlib-js","https://opencollective.com/stdlib","https://tidelift.com/funding/github/npm/@stdlib/stdlib"],"categories":[],"sub_categories":[],"readme":"\u003c!--\n\n@license Apache-2.0\n\nCopyright (c) 2019 The Stdlib Authors.\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n   http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n\n--\u003e\n\n\n\u003cdetails\u003e\n  \u003csummary\u003e\n    About stdlib...\n  \u003c/summary\u003e\n  \u003cp\u003eWe 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.\u003c/p\u003e\n  \u003cp\u003eThe 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.\u003c/p\u003e\n  \u003cp\u003eWhen 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.\u003c/p\u003e\n  \u003cp\u003eTo join us in bringing numerical computing to the web, get started by checking us out on \u003ca href=\"https://github.com/stdlib-js/stdlib\"\u003eGitHub\u003c/a\u003e, and please consider \u003ca href=\"https://opencollective.com/stdlib\"\u003efinancially supporting stdlib\u003c/a\u003e. We greatly appreciate your continued support!\u003c/p\u003e\n\u003c/details\u003e\n\n# ellipj\n\n[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] \u003c!-- [![dependencies][dependencies-image]][dependencies-url] --\u003e\n\n\u003e Compute the [Jacobi elliptic functions][jacobi-elliptic] sn, cn, and dn.\n\n\u003csection class=\"intro\"\u003e\n\nThe [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\n\n\u003c!-- \u003cequation class=\"equation\" label=\"eq:incomplete_elliptic_integral_first_kind\" align=\"center\" raw=\"u=\\int_{0}^{\\varphi}{\\frac {\\mathrm{d} \\theta }{\\sqrt {1-m\\sin^{2}\\theta }}}\" alt=\"Incomplete elliptic integral of the first kind\"\u003e --\u003e\n\n\u003c!-- \u003c/equation\u003e --\u003e\n\nwhere the parameter `m` is related to the modulus `k` by `m = k^2`.\n\n\u003c/section\u003e\n\n\u003c!-- /.intro --\u003e\n\n\u003csection class=\"installation\"\u003e\n\n## Installation\n\n```bash\nnpm install @stdlib/math-base-special-ellipj\n```\n\nAlternatively,\n\n-   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]).\n-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).\n-   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]).\n\nThe [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.\n\nTo 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.\n\n\u003c/section\u003e\n\n\u003csection class=\"usage\"\u003e\n\n## Usage\n\n```javascript\nvar ellipj = require( '@stdlib/math-base-special-ellipj' );\n```\n\n#### ellipj( u, m )\n\nComputes the [Jacobi elliptic functions][jacobi-elliptic] functions `sn`, `cn`, and `dn`, and the Jacobi amplitude `am`.\n\n```javascript\nvar v = ellipj( 0.3, 0.5 );\n// returns [ ~0.293, ~0.956, ~0.978, ~0.298 ]\n\nv = ellipj( 0.0, 0.0 );\n// returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]\n\nv = ellipj( Infinity, 1.0 );\n// returns [ ~1.0, ~0.0, ~0.0, ~1.571 ]\n\nv = ellipj( 0.0, -2.0 );\n// returns [ ~0.0, ~1.0, ~1.0, NaN ]\n\nv = ellipj( NaN, NaN );\n// returns [ NaN, NaN, NaN, NaN ]\n```\n\n#### ellipj.assign( u, m, out, stride, offset )\n\nComputes the Jacobi elliptic functions `sn`, `cn`, `dn`, and Jacobi amplitude `am` and assigns results to a provided output array.\n\n```javascript\nvar Float64Array = require( '@stdlib/array-float64' );\n\nvar out = new Float64Array( 4 );\n\nvar v = ellipj.assign( 0.0, 0.0, out, 1, 0 );\n// returns \u003cFloat64Array\u003e[ ~0.0, ~1.0, ~1.0, ~0.0 ]\n\nvar bool = ( v === out );\n// returns true\n```\n\n#### ellipj.sn( u, m )\n\nComputes the Jacobi elliptic function `sn` of value `u` with modulus `m`.\n\n```javascript\nvar v = ellipj.sn( 0.3, 0.5 );\n// returns ~0.293\n```\n\n#### ellipj.cn( u, m )\n\nComputes the Jacobi elliptic function `cn` of value `u` with modulus `m`.\n\n```javascript\nvar v = ellipj.cn( 0.3, 0.5 );\n// returns ~0.956\n```\n\n#### ellipj.dn( u, m )\n\nComputes the Jacobi elliptic function `dn` of value `u` with modulus `m`.\n\n```javascript\nvar v = ellipj.dn( 0.3, 0.5 );\n// returns ~0.978\n```\n\n#### ellipj.am( u, m )\n\nComputes the Jacobi amplitude `am` of value `u` with modulus `m`.\n\n```javascript\nvar v = ellipj.am( 0.3, 0.5 );\n// returns ~0.298\n\nv = ellipj.am( 0.3, 2.0 );\n// returns NaN\n```\n\nAlthough `sn`, `cn`, and `dn` may be computed for `-∞ \u003c m \u003c ∞`, the domain of `am` is `0 ≤ m ≤ 1`. For `m \u003c 0` or `m \u003e 1`, the function returns `NaN`.\n\n\u003c/section\u003e\n\n\u003c!-- /.usage --\u003e\n\n\u003csection class=\"notes\"\u003e\n\n## Notes\n\n-   Functions `sn`, `cn`, and `dn` are valid for `-∞ \u003c m \u003c ∞`. Values for `m \u003c 0` or `m \u003e 1` are computed in terms of Jacobi elliptic functions with `0 \u003c m \u003c 1` via the transformations outlined in Equations 16.13 and 16.15 from _The Handbook of Mathematical Functions_ (Abramowitz and Stegun).\n-   If more than one of `sn`, `cn`, `dn`, or `am` is to be computed, preferring using `ellipj` to compute all four values simultaneously.\n\n\u003c/section\u003e\n\n\u003c!-- /.notes --\u003e\n\n\u003csection class=\"examples\"\u003e\n\n## Examples\n\n\u003c!-- eslint no-undef: \"error\" --\u003e\n\n```javascript\nvar linspace = require( '@stdlib/array-base-linspace' );\nvar ellipk = require( '@stdlib/math-base-special-ellipk' );\nvar ellipj = require( '@stdlib/math-base-special-ellipj' );\n\nvar m = 0.7;\nvar u = linspace( 0.0, ellipk( m ), 100 );\n\nvar out;\nvar i;\nfor ( i = 0; i \u003c 100; i++ ) {\n    out = ellipj( u[ i ], m );\n    console.log( 'sn(%d, %d) = %d', u[ i ], m, out[ 0 ] );\n    console.log( 'cn(%d, %d) = %d', u[ i ], m, out[ 1 ] );\n    console.log( 'dn(%d, %d) = %d', u[ i ], m, out[ 2 ] );\n    console.log( 'am(%d, %d) = %d', u[ i ], m, out[ 3 ] );\n}\n```\n\n\u003c/section\u003e\n\n\u003c!-- /.examples --\u003e\n\n\u003c!-- C interface documentation. --\u003e\n\n* * *\n\n\u003csection class=\"c\"\u003e\n\n## C APIs\n\n\u003c!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. --\u003e\n\n\u003csection class=\"intro\"\u003e\n\n\u003c/section\u003e\n\n\u003c!-- /.intro --\u003e\n\n\u003c!-- C usage documentation. --\u003e\n\n\u003csection class=\"usage\"\u003e\n\n### Usage\n\n```c\n#include \"stdlib/math/base/special/ellipj.h\"\n```\n\n#### stdlib_base_ellipj( x, m, \u0026sn, \u0026cn, \u0026dn, \u0026am )\n\nComputes the [Jacobi elliptic functions][jacobi-elliptic] functions `sn`, `cn`, and `dn`, and the Jacobi amplitude `am`.\n\n```c\ndouble sn;\ndouble cn;\ndouble dn;\ndouble am;\n\nstdlib_base_ellipj( 0.3, 0.5, \u0026sn, \u0026cn, \u0026dn, \u0026am );\n```\n\nThe function accepts the following arguments:\n\n-   **x**: `[in] double` input value.\n-   **m**: `[in] double` modulus `m`, equivalent to `k²`.\n-   **sn**: `[out] double*` destination for the sine amplitude.\n-   **cn**: `[out] double*` destination for the cosine amplitude.\n-   **dn**: `[out] double*` destination for the delta amplitude.\n-   **am**: `[out] double*` destination for the Jacobi amplitude.\n\n```c\nvoid stdlib_base_ellipj( const double u, const double m, double* sn, double* cn, double* dn, double* am );\n```\n\n\u003c/section\u003e\n\n\u003c!-- /.usage --\u003e\n\n\u003c!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. --\u003e\n\n\u003csection class=\"notes\"\u003e\n\n\u003c/section\u003e\n\n\u003c!-- /.notes --\u003e\n\n\u003c!-- C API usage examples. --\u003e\n\n\u003csection class=\"examples\"\u003e\n\n### Examples\n\n```c\n#include \"stdlib/math/base/special/ellipj.h\"\n#include \u003cstdlib.h\u003e\n#include \u003cstdio.h\u003e\n\nint main( void ) {\n    double sn;\n    double cn;\n    double dn;\n    double am;\n    double x;\n    int i;\n\n    for ( i = 0; i \u003c 100; i++ ) {\n        x = 2.0 * ( (double)rand() / (double)RAND_MAX );\n        stdlib_base_ellipj( x, 0.7, \u0026sn, \u0026cn, \u0026dn, \u0026am );\n        printf( \"x: %lf, m: %lf =\u003e sn: %lf, cn: %lf, dn: %lf, am: %lf\\n\", x, 0.7, sn, cn, dn, am );\n    }\n}\n```\n\n\u003c/section\u003e\n\n\u003c!-- /.examples --\u003e\n\n\u003c/section\u003e\n\n\u003c!-- /.c --\u003e\n\n* * *\n\n\u003csection class=\"references\"\u003e\n\n## References\n\n-   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].\n-   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].\n\n\u003c/section\u003e\n\n\u003c!-- /.references --\u003e\n\n\u003c!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. --\u003e\n\n\u003csection class=\"related\"\u003e\n\n* * *\n\n## See Also\n\n-   \u003cspan class=\"package-name\"\u003e[`@stdlib/math-base/special/ellipe`][@stdlib/math/base/special/ellipe]\u003c/span\u003e\u003cspan class=\"delimiter\"\u003e: \u003c/span\u003e\u003cspan class=\"description\"\u003ecompute the complete elliptic integral of the second kind.\u003c/span\u003e\n-   \u003cspan class=\"package-name\"\u003e[`@stdlib/math-base/special/ellipk`][@stdlib/math/base/special/ellipk]\u003c/span\u003e\u003cspan class=\"delimiter\"\u003e: \u003c/span\u003e\u003cspan class=\"description\"\u003ecompute the complete elliptic integral of the first kind.\u003c/span\u003e\n\n\u003c/section\u003e\n\n\u003c!-- /.related --\u003e\n\n\u003c!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --\u003e\n\n\n\u003csection class=\"main-repo\" \u003e\n\n* * *\n\n## Notice\n\nThis 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.\n\nFor 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].\n\n#### Community\n\n[![Chat][chat-image]][chat-url]\n\n---\n\n## License\n\nSee [LICENSE][stdlib-license].\n\n\n## Copyright\n\nCopyright \u0026copy; 2016-2025. The Stdlib [Authors][stdlib-authors].\n\n\u003c/section\u003e\n\n\u003c!-- /.stdlib --\u003e\n\n\u003c!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --\u003e\n\n\u003csection class=\"links\"\u003e\n\n[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-ellipj.svg\n[npm-url]: https://npmjs.org/package/@stdlib/math-base-special-ellipj\n\n[test-image]: https://github.com/stdlib-js/math-base-special-ellipj/actions/workflows/test.yml/badge.svg?branch=main\n[test-url]: https://github.com/stdlib-js/math-base-special-ellipj/actions/workflows/test.yml?query=branch:main\n\n[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-ellipj/main.svg\n[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-ellipj?branch=main\n\n\u003c!--\n\n[dependencies-image]: https://img.shields.io/david/stdlib-js/math-base-special-ellipj.svg\n[dependencies-url]: https://david-dm.org/stdlib-js/math-base-special-ellipj/main\n\n--\u003e\n\n[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg\n[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im\n\n[stdlib]: https://github.com/stdlib-js/stdlib\n\n[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors\n\n[umd]: https://github.com/umdjs/umd\n[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules\n\n[deno-url]: https://github.com/stdlib-js/math-base-special-ellipj/tree/deno\n[deno-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/deno/README.md\n[umd-url]: https://github.com/stdlib-js/math-base-special-ellipj/tree/umd\n[umd-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/umd/README.md\n[esm-url]: https://github.com/stdlib-js/math-base-special-ellipj/tree/esm\n[esm-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/esm/README.md\n[branches-url]: https://github.com/stdlib-js/math-base-special-ellipj/blob/main/branches.md\n\n[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-ellipj/main/LICENSE\n\n[jacobi-elliptic]: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions\n\n[incomplete-elliptic]: https://en.wikipedia.org/wiki/Elliptic_integral\n\n[@fukushima:2009a]: https://doi.org/10.1007/s10569-009-9228-z\n\n[@fukushima:2015a]: https://doi.org/10.1016/j.cam.2014.12.038\n\n\u003c!-- \u003crelated-links\u003e --\u003e\n\n[@stdlib/math/base/special/ellipe]: https://github.com/stdlib-js/math-base-special-ellipe\n\n[@stdlib/math/base/special/ellipk]: https://github.com/stdlib-js/math-base-special-ellipk\n\n\u003c!-- \u003c/related-links\u003e --\u003e\n\n\u003c/section\u003e\n\n\u003c!-- /.links --\u003e\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fstdlib-js%2Fmath-base-special-ellipj","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fstdlib-js%2Fmath-base-special-ellipj","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fstdlib-js%2Fmath-base-special-ellipj/lists"}