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feat: add math/base/special/hyp2f1
PR-URL: #5140 Closes: #122 Closes: #216 Reviewed-by: Athan Reines <[email protected]> Co-authored-by: stdlib-bot <[email protected]>
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Diff for: ‎lib/node_modules/@stdlib/math/base/special/hyp2f1/README.md

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<!--
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@license Apache-2.0
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Copyright (c) 2025 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# Gaussian hypergeometric function
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> Evaluates the [Gaussian hypergeometric function][hypergeometric-function].
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<section class="intro">
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The [Gaussian hypergeometric function][hypergeometric-function] is defined for `|x| < 1` by the power series:
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<!-- <equation class="equation" label="eq:hypergeometric_function" align="center" raw="{}_2F_1(a, b; c; x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{x^n}{n!} = 1 + \frac{a b}{c} x + \frac{a(a+1) b(b+1)}{c(c+1)} \frac{x^2}{2!} + \frac{a(a+1)(a+2) b(b+1)(b+2)}{c(c+1)(c+2)} \frac{x^3}{3!} + \cdots" alt="Gaussian hypergeometric function."> -->
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```math
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{}_2F_1(a, b; c; x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{x^n}{n!} = 1 + \frac{a b}{c} x + \frac{a(a+1) b(b+1)}{c(c+1)} \frac{x^2}{2!} + \frac{a(a+1)(a+2) b(b+1)(b+2)}{c(c+1)(c+2)} \frac{x^3}{3!} + \cdots
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```
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<!-- <div class="equation" align="center" data-raw-text="{}_2F_1(a, b; c; x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{x^n}{n!} = 1 + \frac{a b}{c} x + \frac{a(a+1) b(b+1)}{c(c+1)} \frac{x^2}{2!} + \frac{a(a+1)(a+2) b(b+1)(b+2)}{c(c+1)(c+2)} \frac{x^3}{3!} + \cdots" data-equation="eq:hypergeometric_function">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/hyp2f1/docs/img/equation_hypergeometric_function.svg" alt="Gaussian hypergeometric function.">
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<br>
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</div> -->
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<!-- </equation> -->
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and is undefined (or infinite) if `c` equals a non-positive integer.
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Here `(q)ₙ` is the (rising) [Pochhammer symbol][pochhammer-symbol], which is defined by:
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<!-- <equation class="equation" label="eq:pochhammer_symbol" align="center" raw="(q)_n = \begin{cases} 1 & n = 0 \\ q(q+1) \cdots (q+n-1) & n > 0 \end{cases}" alt="Pochhammer symbol."> -->
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```math
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(q)_n = \begin{cases} 1 & n = 0 \\ q(q+1) \cdots (q+n-1) & n > 0 \end{cases}
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```
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<!-- <div class="equation" align="center" data-raw-text="(q)_n = \begin{cases} 1 & n = 0 \\ q(q+1) \cdots (q+n-1) & n > 0 \end{cases}" data-equation="eq:pochhammer_symbol">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/hyp2f1/docs/img/equation_pochhammer_symbol.svg" alt="Pochhammer symbol.">
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<br>
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</div> -->
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<!-- </equation> -->
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For `|x| >= 1`, the function can be [analytically continued][analytic-continuation] using functional identities and transformation formulas.
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var hyp2f1 = require( '@stdlib/math/base/special/hyp2f1' );
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```
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#### hyp2f1( a, b, c, x )
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Evaluates the [Gaussian hypergeometric function][hypergeometric-function].
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```javascript
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var v = hyp2f1( 1.0, 1.0, 1.0, 0.0 );
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// returns 1.0
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v = hyp2f1( 10.0, 7.4, -1.8, -0.99 );
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// returns ~0.423
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v = hyp2f1( 3.0, 4.0, 7.0, 1.0 );
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// returns +Infinity
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v = hyp2f1( NaN, 3.0, 2.0, 0.5 );
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// returns NaN
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var linspace = require( '@stdlib/array/base/linspace' );
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var hyp2f1 = require( '@stdlib/math/base/special/hyp2f1' );
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var a = linspace( -50.0, 50.0, 100 );
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var b = linspace( -50.0, 50.0, 100 );
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var c = linspace( -50.0, 50.0, 100 );
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var x = linspace( -50.0, 50.0, 100 );
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var i;
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for ( i = 0; i < x.length; i++ ) {
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console.log( 'a: %d, b: %d, c: %d, x: %d, 2F1(a,b;c;x): %d', a[ i ], b[ i ], c[ i ], x[ i ], hyp2f1( a[ i ], b[ i ], c[ i ], x[ i ] ) );
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}
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```
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</section>
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<!-- /.examples -->
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<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
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<section class="related">
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</section>
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<!-- /.related -->
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<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="links">
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[hypergeometric-function]: https://en.wikipedia.org/wiki/Hypergeometric_function
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[pochhammer-symbol]: https://en.wikipedia.org/wiki/Falling_and_rising_factorials
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[analytic-continuation]: https://en.wikipedia.org/wiki/Analytic_continuation
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</section>
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<!-- /.links -->

Diff for: ‎lib/node_modules/@stdlib/math/base/special/hyp2f1/benchmark/benchmark.js

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/**
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* @license Apache-2.0
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*
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* Copyright (c) 2025 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var uniform = require( '@stdlib/random/array/uniform' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var pkg = require( './../package.json' ).name;
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var hyp2f1 = require( './../lib' );
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// MAIN //
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bench( pkg, function benchmark( bm ) {
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var x;
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var a;
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var b;
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var c;
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var z;
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var i;
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a = uniform( 100, -50.0, 50.0 );
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b = uniform( 100, -50.0, 50.0 );
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c = uniform( 100, -50.0, 50.0 );
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x = uniform( 100, -50.0, 50.0 );
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bm.tic();
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for ( i = 0; i < bm.iterations; i++ ) {
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// eslint-disable-next-line max-len
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z = hyp2f1( a[ i % x.length ], b[ i % x.length ], c[ i % x.length ], x[ i % x.length ] );
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if ( isnan( z ) ) {
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bm.fail( 'should not return NaN' );
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}
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}
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bm.toc();
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if ( isnan( z ) ) {
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bm.fail( 'should not return NaN' );
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}
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bm.pass( 'benchmark finished' );
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bm.end();
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});

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