106 lines
2.4 KiB
Markdown
106 lines
2.4 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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# fresnels
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> Compute the [Fresnel integral][fresnel-integral] S(x).
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<section class="intro">
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The [Fresnel integral][fresnel-integral] S(x) is defined as
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<!-- <equation class="equation" label="eq:fresnel_integral" align="center" raw="S(x) = \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t." alt="Fresnel integral S(x)"> -->
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<div class="equation" align="center" data-raw-text="S(x) = \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t." data-equation="eq:fresnel_integral">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/fresnels/docs/img/equation_fresnel_integral.svg" alt="Fresnel integral S(x)">
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<br>
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</div>
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<!-- </equation> -->
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Some sources define S(x) using t<sup>2</sup> for the argument of the sine. To get this function, multiply the computed integral by `√(π/2)` and multiply the argument `x` by `√(2/π)`.
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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 fresnels = require( '@stdlib/math/base/special/fresnels' );
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```
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#### fresnels( x )
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Computes the [Fresnel integral][fresnel-integral] S(x).
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```javascript
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var v = fresnels( 0.0 );
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// returns ~0.0
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v = fresnels( 1.0 );
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// returns ~0.438
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v = fresnels( Infinity );
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// returns ~0.5
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v = fresnels( -Infinity );
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// returns ~-0.5
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v = fresnels( 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 fresnels = require( '@stdlib/math/base/special/fresnels' );
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var x = linspace( 0.0, 10.0, 100 );
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var i;
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for ( i = 0; i < x.length; i++ ) {
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console.log( fresnels( 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 class="links">
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[fresnel-integral]: https://en.wikipedia.org/wiki/Fresnel_integral
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</section>
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<!-- /.links -->
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