npm package: @stdlib/math-base-special-fresnel
Description: Compute the Fresnel integrals S(x) and C(x).
Author: The Stdlib Authors
Latest version: 0.3.1

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# fresnel

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> 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="installation">

## Installation

```bash
npm install @stdlib/math-base-special-fresnel
```

</section>

<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 -->

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

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---

## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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

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[@stdlib/math/base/special/fresnelc]: https://www.npmjs.com/package/@stdlib/math-base-special-fresnelc

[@stdlib/math/base/special/fresnels]: https://www.npmjs.com/package/@stdlib/math-base-special-fresnels

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</section>

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