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Stdlib Authors","url":"https://github.com/stdlib-js/stdlib/graphs/contributors"},"license":"Apache-2.0 AND BSL-1.0","homepage":"https://stdlib.io","keywords":["stdlib","stdmath","mathematics","math","special function","special","function","gamma","lanczos","scaled","approx","approximation","polynomial","rational","factorial","number"],"repository":{"type":"git","url":"git://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled.git"},"description":"Calculate a scaled Lanczos sum for the approximation of the gamma function.","contributors":[{"name":"The Stdlib Authors","url":"https://github.com/stdlib-js/stdlib/graphs/contributors"}],"maintainers":[{"name":"stdlib-bot","email":"kgryte+stdlibbot@gmail.com"},{"name":"kgryte","email":"kgryte@gmail.com"},{"name":"planeshifter","email":"pgb@andrew.cmu.edu"},{"name":"rreusser","email":"rsreusser@gmail.com"}],"readme":"<!--\n\n@license Apache-2.0\n\nCopyright (c) 2018 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-->\n\n\n<details>\n  <summary>\n    About stdlib...\n  </summary>\n  <p>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.</p>\n  <p>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.</p>\n  <p>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.</p>\n  <p>To join us in bringing numerical computing to the web, get started by checking us out on <a href=\"https://github.com/stdlib-js/stdlib\">GitHub</a>, and please consider <a href=\"https://opencollective.com/stdlib\">financially supporting stdlib</a>. We greatly appreciate your continued support!</p>\n</details>\n\n# Gamma Scaled Lanczos Sum\n\n[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->\n\n> Calculate a scaled Lanczos sum for the approximation of the [gamma function][gamma-function].\n\n<section class=\"intro\">\n\nThe [Lanczos approximation][lanczos-approximation] for the [gamma function][gamma-function] can be written in partial fraction form as follows:\n\n<!-- <equation class=\"equation\" label=\"eq:lanczos_approximation\" align=\"center\" raw=\"\\Gamma ( n ) = \\frac{(n+g-0.5)^{n-0.5}}{e^{n+g-0.5}} L_g(n)\" alt=\"Lanczos approximation for gamma function.\"> -->\n\n<div class=\"equation\" align=\"center\" data-raw-text=\"\\Gamma ( n ) = \\frac{(n+g-0.5)^{n-0.5}}{e^{n+g-0.5}} L_g(n)\" data-equation=\"eq:lanczos_approximation\">\n    <img src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled/docs/img/equation_lanczos_approximation.svg\" alt=\"Lanczos approximation for gamma function.\">\n    <br>\n</div> -->\n\n<!-- </equation> -->\n\nwhere `g` is an [arbitrary constant][@stdlib/constants/float64/gamma-lanczos-g] and `L_g(n)` is the Lanczos sum. The scaled Lanczos sum is given by\n\n<!-- <equation class=\"equation\" label=\"eq:scaled_lanczos_sum\" align=\"center\" raw=\"L_g(n) \\cdot \\exp(-g)\" alt=\"Scaled Lanczos sum.\"> -->\n\n<!-- <div class=\"equation\" align=\"center\" data-raw-text=\"L_g(n) \\cdot \\exp(-g)\" data-equation=\"eq:scaled_lanczos_sum\">\n    <img src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled/docs/img/equation_scaled_lanczos_sum.svg\" alt=\"Scaled Lanczos sum.\">\n    <br>\n</div>\n\n<!-- </equation> -->\n\n</section>\n\n<!-- /.intro -->\n\n<section class=\"installation\">\n\n## Installation\n\n```bash\nnpm install @stdlib/math-base-special-gamma-lanczos-sum-expg-scaled\n```\n\n</section>\n\n<section class=\"usage\">\n\n## Usage\n\n```javascript\nvar gammaLanczosSumExpGScaled = require( '@stdlib/math-base-special-gamma-lanczos-sum-expg-scaled' );\n```\n\n#### gammaLanczosSumExpGScaled( x )\n\nCalculates the Lanczos sum for the approximation of the [gamma function][gamma-function] (scaled by `exp(-g)`, where `g = 10.900511`).\n\n```javascript\nvar v = gammaLanczosSumExpGScaled( 4.0 );\n// returns ~0.018\n\nv = gammaLanczosSumExpGScaled( -1.5 );\n// returns ~25.337\n\nv = gammaLanczosSumExpGScaled( -0.5 );\n// returns ~-12.911\n\nv = gammaLanczosSumExpGScaled( 0.5 );\n// returns ~1.772\n\nv = gammaLanczosSumExpGScaled( 0.0 );\n// returns Infinity\n\nv = gammaLanczosSumExpGScaled( NaN );\n// returns NaN\n```\n\n</section>\n\n<!-- /.usage -->\n\n<section class=\"examples\">\n\n## Examples\n\n<!-- eslint no-undef: \"error\" -->\n\n```javascript\nvar uniform = require( '@stdlib/random-array-uniform' );\nvar logEachMap = require( '@stdlib/console-log-each-map' );\nvar gammaLanczosSumExpGScaled = require( '@stdlib/math-base-special-gamma-lanczos-sum-expg-scaled' );\n\nvar opts = {\n    'dtype': 'float64'\n};\nvar x = uniform( 100, -10.0, 10.0, opts );\n\nlogEachMap( 'x: %0.4f, f(x): %0.4f', x, gammaLanczosSumExpGScaled );\n```\n\n</section>\n\n<!-- /.examples -->\n\n<!-- C interface documentation. -->\n\n* * *\n\n<section class=\"c\">\n\n## C APIs\n\n<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->\n\n<section class=\"intro\">\n\n</section>\n\n<!-- /.intro -->\n\n<!-- C usage documentation. -->\n\n<section class=\"usage\">\n\n### Usage\n\n```c\n#include \"stdlib/math/base/special/gamma_lanczos_sum_expg_scaled.h\"\n```\n\n#### stdlib_base_gamma_lanczos_sum_expg_scaled( x )\n\nCalculates the Lanczos sum for the approximation of the [gamma function][gamma-function] (scaled by `exp(-g)`, where `g = 10.900511`).\n\n```c\ndouble out = stdlib_base_gamma_lanczos_sum_expg_scaled( 4.0 );\n// returns ~0.018\n\nout = stdlib_base_gamma_lanczos_sum_expg_scaled( -1.5 );\n// returns ~25.337\n```\n\nThe function accepts the following arguments:\n\n-   **x**: `[in] double` input value.\n\n```c\ndouble stdlib_base_gamma_lanczos_sum_expg_scaled( const double x );\n```\n\n</section>\n\n<!-- /.usage -->\n\n<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->\n\n<section class=\"notes\">\n\n</section>\n\n<!-- /.notes -->\n\n<!-- C API usage examples. -->\n\n<section class=\"examples\">\n\n### Examples\n\n```c\n#include \"stdlib/math/base/special/gamma_lanczos_sum_expg_scaled.h\"\n#include <stdlib.h>\n#include <stdio.h>\n\nint main( void ) {\n    const double x[] = { 4.0, -1.5, -0.5, 0.5 };\n\n    double y;\n    int i;\n    for ( i = 0; i < 4; i++ ) {\n        y = stdlib_base_gamma_lanczos_sum_expg_scaled( x[ i ] );\n        printf( \"gamma_lanczos_sum_expg_scaled(%lf) = %lf\\n\", x[ i ], y );\n    }\n}\n```\n\n</section>\n\n<!-- /.examples -->\n\n</section>\n\n<!-- /.c -->\n\n<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->\n\n<section class=\"related\">\n\n* * *\n\n## See Also\n\n-   <span class=\"package-name\">[`@stdlib/math-base/special/gamma`][@stdlib/math/base/special/gamma]</span><span class=\"delimiter\">: </span><span class=\"description\">gamma function.</span>\n-   <span class=\"package-name\">[`@stdlib/math-base/special/gamma-lanczos-sum`][@stdlib/math/base/special/gamma-lanczos-sum]</span><span class=\"delimiter\">: </span><span class=\"description\">calculate the Lanczos sum for the approximation of the gamma function.</span>\n\n</section>\n\n<!-- /.related -->\n\n<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->\n\n\n<section class=\"main-repo\" >\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## Copyright\n\nCopyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].\n\n</section>\n\n<!-- /.stdlib -->\n\n<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->\n\n<section class=\"links\">\n\n[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-gamma-lanczos-sum-expg-scaled.svg\n[npm-url]: https://npmjs.org/package/@stdlib/math-base-special-gamma-lanczos-sum-expg-scaled\n\n[test-image]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/actions/workflows/test.yml/badge.svg?branch=v0.2.2\n[test-url]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/actions/workflows/test.yml?query=branch:v0.2.2\n\n[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/main.svg\n[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled?branch=main\n\n<!--\n\n[dependencies-image]: https://img.shields.io/david/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled.svg\n[dependencies-url]: https://david-dm.org/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/main\n\n-->\n\n[chat-image]: https://img.shields.io/badge/zulip-join_chat-brightgreen.svg\n[chat-url]: https://stdlib.zulipchat.com\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-gamma-lanczos-sum-expg-scaled/tree/deno\n[deno-readme]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/blob/deno/README.md\n[umd-url]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/tree/umd\n[umd-readme]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/blob/umd/README.md\n[esm-url]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/tree/esm\n[esm-readme]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/blob/esm/README.md\n[branches-url]: https://github.com/stdlib-js/math-base-special-gamma-lanczos-sum-expg-scaled/blob/main/branches.md\n\n[@stdlib/constants/float64/gamma-lanczos-g]: https://www.npmjs.com/package/@stdlib/constants-float64-gamma-lanczos-g\n\n[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function\n\n[lanczos-approximation]: https://en.wikipedia.org/wiki/Lanczos_approximation\n\n<!-- <related-links> -->\n\n[@stdlib/math/base/special/gamma]: https://www.npmjs.com/package/@stdlib/math-base-special-gamma\n\n[@stdlib/math/base/special/gamma-lanczos-sum]: https://www.npmjs.com/package/@stdlib/math-base-special-gamma-lanczos-sum\n\n<!-- </related-links> -->\n\n</section>\n\n<!-- /.links -->\n","readmeFilename":"README.md"}