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# fresnels

> Compute the [Fresnel integral][fresnel-integral] S(x).

<section class="intro">

The [Fresnel integral][fresnel-integral] S(x) is defined as

<!-- <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)"> -->

<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">
    <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)">
    <br>
</div>

<!-- </equation> -->

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)`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var fresnels = require( '@stdlib/math/base/special/fresnels' );
```

#### fresnels( x )

Computes the [Fresnel integral][fresnel-integral] S(x).

```javascript
var v = fresnels( 0.0 );
// returns ~0.0

v = fresnels( 1.0 );
// returns ~0.438

v = fresnels( Infinity );
// returns ~0.5

v = fresnels( -Infinity );
// returns ~-0.5

v = fresnels( NaN );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var linspace = require( '@stdlib/array/linspace' );
var fresnels = require( '@stdlib/math/base/special/fresnels' );

var x = linspace( 0.0, 10.0, 100 );
var i;

for ( i = 0; i < x.length; i++ ) {
    console.log( fresnels( x[ i ] ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

[fresnel-integral]: https://en.wikipedia.org/wiki/Fresnel_integral

</section>

<!-- /.links -->