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# fresnel

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

<section class="intro">

The [Fresnel integrals][fresnel-integral] are defined as

<!-- <equation class="equation" label="eq:fresnel_integrals" align="center" raw="\begin{align} S(x) &= \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t, \\ C(x) &= \int_0^x \cos\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t. \end{align}" alt="Fresnel integral"> -->

<div class="equation" align="center" data-raw-text="\begin{align} S(x) &amp;= \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t, \\ C(x) &amp;= \int_0^x \cos\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t. \end{align}" data-equation="eq:fresnel_integrals">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/fresnel/docs/img/equation_fresnel_integrals.svg" alt="Fresnel integral">
    <br>
</div>

<!-- </equation> -->

Some sources define the Fresnel integrals using t<sup>2</sup> for the argument of the sine and cosine. To get these functions, multiply the computed integrals by `√(π/2)` and multiply the argument `x` by `√(2/π)`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### fresnel( x )

Simultaneously computes the [Fresnel integrals][fresnel-integral] S(x) and C(x).

```javascript
var v = fresnel( 0.0 );
// returns [ ~0.0, ~0.0 ]

v = fresnel( 1.0 );
// returns [ ~0.438, ~0.780 ]

v = fresnel( Infinity );
// returns [ ~0.5, ~0.5 ]

v = fresnel( -Infinity );
// returns [ ~-0.5, ~-0.5 ]

v = fresnel( NaN );
// returns [ NaN, NaN ]
```

#### fresnel.assign( x, out, stride, offset )

Simultaneously computes the [Fresnel integrals][fresnel-integral] S(x) and C(x) and assigns results to a provided output array.

```javascript
var Float64Array = require( '@stdlib/array/float64' );

var out = new Float64Array( 2 );

var v = fresnel.assign( 0.0, out, 1, 0 );
// returns <Float64Array>[ ~0.0, ~0.0 ]

var bool = ( v === out );
// returns true
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

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

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

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

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/math/base/special/fresnel.h"
```

#### stdlib_base_fresnel( x, &S, &C )

Simultaneously computes the [Fresnel integrals][fresnel-integral] S(x) and C(x).

```c
double S;
double C;

stdlib_base_fresnel( 4.0, &S, &C );
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **S**: `[out] double*` destination for S(x).
-   **C**: `[out] double*` destination for C(x).

```c
void stdlib_base_fresnel( const double x, double *S, double *C );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/math/base/special/fresnel.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 0.0, 1.57, 3.14, 6.28 };

    double S;
    double C;
    int i;
    for ( i = 0; i < 4; i++ ) {
        stdlib_base_fresnel( x[ i ], &S, &C );
        printf( "x: %lf => S(x): %lf, C(x): %lf\n", x[ i ], S, C );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math/base/special/fresnelc`][@stdlib/math/base/special/fresnelc]</span><span class="delimiter">: </span><span class="description">compute the Fresnel integral C(x).</span>
-   <span class="package-name">[`@stdlib/math/base/special/fresnels`][@stdlib/math/base/special/fresnels]</span><span class="delimiter">: </span><span class="description">compute the Fresnel integral S(x).</span>

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

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

<!-- <related-links> -->

[@stdlib/math/base/special/fresnelc]: https://github.com/stdlib-js/math/tree/main/base/special/fresnelc

[@stdlib/math/base/special/fresnels]: https://github.com/stdlib-js/math/tree/main/base/special/fresnels

<!-- </related-links> -->

</section>

<!-- /.links -->
