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

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# fresnels

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> 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="installation">

## Installation

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

</section>

<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-base-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 -->

<!-- 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/fresnels.h"
```

#### stdlib_base_fresnels( x )

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

```c
double out = stdlib_base_fresnels( 0.0 );
// returns ~0.0

out = stdlib_base_fresnels( 1.0 );
// returns ~0.438
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.

```c
double stdlib_base_fresnels( const double x );
```

</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/fresnels.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 0.0, 3.14, 5.55, 10.0 };

    double y;
    int i;
    for ( i = 0; i < 4; i++ ) {
        y = stdlib_base_fresnels( x[ i ] );
        printf( "fresnels(%lf) = %lf\n", x[ i ], y );
    }
}
```

</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/fresnel`][@stdlib/math/base/special/fresnel]</span><span class="delimiter">: </span><span class="description">compute the Fresnel integrals S(x) and C(x).</span>
-   <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>

</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].

</section>

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[branches-url]: https://github.com/stdlib-js/math-base-special-fresnels/blob/main/branches.md

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

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

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

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

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</section>

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