119 lines
3.2 KiB
Markdown
119 lines
3.2 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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# betaln
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> [Natural logarithm][natural-logarithm] of the [beta function][beta-function].
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<section class="intro">
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The [beta function][beta-function], also called the Euler integral, is defined as
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<!-- <equation class="equation" label="eq:beta_function" align="center" raw="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" alt="Equation for the beta function."> -->
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<div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" data-equation="eq:beta_function">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/betaln/docs/img/equation_beta_function.svg" alt="Equation for the beta function.">
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<br>
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</div>
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<!-- </equation> -->
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The [beta function][beta-function] is related to the [gamma function][gamma-function] via the following equation
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<!-- <equation class="equation" label="eq:beta_function2" align="center" raw="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" alt="Beta function expressed in terms of the Gamma function."> -->
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<div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" data-equation="eq:beta_function2">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/betaln/docs/img/equation_beta_function2.svg" alt="Beta function expressed in terms of the Gamma function.">
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<br>
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</div>
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<!-- </equation> -->
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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 betaln = require( '@stdlib/math/base/special/betaln' );
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```
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#### betaln( x, y )
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Evaluates the the [natural logarithm][natural-logarithm] of the [beta function][beta-function].
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```javascript
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var val = betaln( 0.0, 0.0 );
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// returns Infinity
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val = betaln( 1.0, 1.0 );
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// returns 0.0
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val = betaln( -1.0, 2.0 );
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// returns NaN
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val = betaln( 5.0, 0.2 );
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// returns ~1.218
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val = betaln( 4.0, 1.0 );
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// returns ~-1.386
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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 betaln = require( '@stdlib/math/base/special/betaln' );
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var x;
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var y;
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for ( x = 0; x < 10; x++ ) {
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for ( y = 10; y > 0; y-- ) {
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console.log( 'x: %d, \t y: %d, \t f(x,y): %d', x, y, betaln( x, y ) );
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}
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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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[natural-logarithm]: https://en.wikipedia.org/wiki/Natural_logarithm
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[beta-function]: http://en.wikipedia.org/wiki/Beta_function
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[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function
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</section>
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<!-- /.links -->
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