115 lines
3.0 KiB
Markdown
115 lines
3.0 KiB
Markdown
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@license Apache-2.0
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Copyright (c) 2018 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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# Gamma Lanczos Sum
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> Calculate the Lanczos sum for the approximation of the [gamma function][gamma-function].
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<section class="intro">
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The [Lanczos approximation][lanczos-approximation] for the [gamma function][gamma-function] can be written in partial fraction form as follows:
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<!-- <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."> -->
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<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">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma-lanczos-sum/docs/img/equation_lanczos_approximation.svg" alt="Lanczos approximation for gamma function.">
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<br>
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</div>
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<!-- </equation> -->
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where `g` is an [arbitrary constant][@stdlib/constants/float64/gamma-lanczos-g] and `L_g(n)` is the Lanczos sum.
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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 gammaLanczosSum = require( '@stdlib/math/base/special/gamma-lanczos-sum' );
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```
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#### gammaLanczosSum( x )
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Calculates the Lanczos sum for the approximation of the [gamma function][gamma-function].
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```javascript
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var v = gammaLanczosSum( 4.0 );
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// returns ~950.366
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v = gammaLanczosSum( -1.5 );
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// returns ~1373366.245
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v = gammaLanczosSum( -0.5 );
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// returns ~-699841.735
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v = gammaLanczosSum( 0.5 );
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// returns ~96074.186
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v = gammaLanczosSum( 0.0 );
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// returns Infinity
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v = gammaLanczosSum( NaN );
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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/linspace' );
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var gammaLanczosSum = require( '@stdlib/math/base/special/gamma-lanczos-sum' );
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var x = linspace( -10.0, 10.0, 100 );
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var v;
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var i;
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for ( i = 0; i < x.length; i++ ) {
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v = gammaLanczosSum( x[ i ] );
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console.log( 'x: %d, f(x): %d', x[ i ], v );
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}
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[@stdlib/constants/float64/gamma-lanczos-g]: https://www.npmjs.com/package/@stdlib/constants-float64-gamma-lanczos-g
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[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function
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[lanczos-approximation]: https://en.wikipedia.org/wiki/Lanczos_approximation
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</section>
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<!-- /.links -->
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