y0

Compute the Bessel function of the second kind of order zero.

The Bessel function of the second kind of order zero is defined as

$$Y_0(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta) \, d\theta -\frac{2}{\pi} \int_0^\infty e^{-x \sinh t} \, dt.$$

Usage

var y0 = require( '@stdlib/math/base/special/bessely0' );

y0( x )

Computes the Bessel function of the second kind of order zero at x.

var v = y0( 0.0 );
// returns -Infinity

v = y0( 1.0 );
// returns ~0.088

v = y0( Infinity );
// returns 0.0

If x < 0 or x is NaN, the function returns NaN.

var v = y0( -1.0 );
// returns NaN

v = y0( -Infinity );
// returns NaN

v = y0( NaN );
// returns NaN

Examples

var uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var bessely0 = require( '@stdlib/math/base/special/bessely0' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, 0.0, 100.0, opts );

logEachMap( 'bessely0(%0.4f) = %0.4f', x, bessely0 );

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C APIs

Usage

#include "stdlib/math/base/special/bessely0.h"

stdlib_base_bessely0( x )

Computes the Bessel function of the second kind of order zero at x.

double out = stdlib_base_bessely0( 0.0 );
// returns -Infinity

out = stdlib_base_bessely0( 1.0 );
// returns ~0.088

The function accepts the following arguments:

• x: [in] double input value.

double stdlib_base_bessely0( const double x );

Examples

#include "stdlib/math/base/special/bessely0.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 0.0, 1.0, 2.0, 3.0, 4.0 };

    double y;
    int i;
    for ( i = 0; i < 5; i++ ) {
        y = stdlib_base_bessely0( x[ i ] );
        printf( "bessely0(%lf) = %lf\n", x[ i ], y );
    }
}

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See Also

• @stdlib/math/base/special/besselj0: compute the Bessel function of the first kind of order zero.
• @stdlib/math/base/special/besselj1: compute the Bessel function of the first kind of order one.
• @stdlib/math/base/special/bessely1: compute the Bessel function of the second kind of order one.