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fresnel

Compute the Fresnel integrals S(x) and C(x).

The Fresnel integrals are defined as

$$\begin{align} S(x) &= \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t, \\ C(x) &= \int_0^x \cos\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t. \end{align}$$

Some sources define the Fresnel integrals using t² 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).

Installation

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

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

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

fresnel( x )

Simultaneously computes the Fresnel integrals S(x) and C(x).

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 S(x) and C(x) and assigns results to a provided output array.

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

Examples

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 ] ) );
}

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C APIs

Usage

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

stdlib_base_fresnel( x, &S, &C )

Simultaneously computes the Fresnel integrals S(x) and C(x).

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).

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

Examples

#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 );
    }
}

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See Also

• @stdlib/math-base/special/fresnelc: compute the Fresnel integral C(x).
• @stdlib/math-base/special/fresnels: compute the Fresnel integral S(x).

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Notice

This package is part of 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, see the main project repository.

Community

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Copyright

Copyright © 2016-2024. The Stdlib Authors.