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* @license Apache-2.0
*
* Copyright (c) 2026 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*
* ## Notice
*
* The original C++ code and copyright notice are from the [Boost library]{@link https://www.boost.org/doc/libs/1_85_0/boost/math/special_functions/detail/bessel_k0.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* Copyright Xiaogang Zhang, 2006.
* Copyright (c) 2017 John Maddock
*
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
* ```
*/
'use strict';
// MODULES //
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var exp = require( '@stdlib/math/base/special/exp' );
var ln = require( '@stdlib/math/base/special/ln' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var r1 = require( './rational_p1q1.js' );
var p2 = require( './poly_p2.js' );
var r3 = require( './rational_p3q3.js' );
// VARIABLES //
var Y_SMALL = 1.137250900268554688;
var LOG_MAX = 709.782712893384;
// MAIN //
/**
* Computes the modified Bessel function of the second kind of order zero.
*
* ## Notes
*
* - Accuracy for subnormal `x` is very poor. Full accuracy is achieved at approximately `1.0e-308` but degrades rapidly as `x → 0`.
*
* @param {number} x - input value
* @returns {number} evaluated Bessel function
*
* @example
* var v = k0( 0.0 );
* // returns Infinity
*
* @example
* var v = k0( 1.0 );
* // returns ~0.421
*
* @example
* var v = k0( 5.0 );
* // returns ~0.00369
*
* @example
* var v = k0( -1.0 );
* // returns NaN
*
* @example
* var v = k0( Infinity );
* // returns 0.0
*/
function k0( x ) {
var ex;
var a;
if ( x < 0.0 ) {
return NaN;
}
if ( x === 0.0 ) {
return PINF;
}
if ( x === PINF ) {
return 0.0;
}
// x <= 1
if ( x <= 1.0 ) {
a = ( x * x ) * 0.25;
a = ( ( r1( a ) + Y_SMALL ) * a ) + 1.0;
return p2( x * x ) - ( ln( x ) * a );
}
// x > 1
if ( x < LOG_MAX ) {
return ( r3( 1.0 / x ) + 1.0 ) *
exp( -x ) / sqrt( x );
}
// overflow-safe
ex = exp( -x * 0.5 );
return ( r3( 1.0 / x ) + 1.0 ) *
ex / sqrt( x ) * ex;
}
// EXPORTS //
module.exports = k0;
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