112 lines
2.6 KiB
Markdown
112 lines
2.6 KiB
Markdown
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@license Apache-2.0
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Copyright (c) 2018 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# y0
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> Compute the [Bessel function of the second kind][bessel-second-kind] of order zero.
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<section class="intro">
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The [Bessel function of the second kind][bessel-second-kind] of order zero is defined as
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<!-- <equation class="equation" label="eq:bessel_second_kind_order_zero" align="center" raw="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." alt="Bessel function of the second kind of order zero"> -->
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<div class="equation" align="center" data-raw-text="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." data-equation="eq:bessel_second_kind_order_zero">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/bessely0/docs/img/equation_bessel_second_kind_order_zero.svg" alt="Bessel function of the second kind of order zero">
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<br>
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</div>
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<!-- </equation> -->
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var y0 = require( '@stdlib/math/base/special/bessely0' );
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```
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#### y0( x )
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Computes the [Bessel function of the second kind][bessel-second-kind] of order zero at `x`.
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```javascript
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var v = y0( 0.0 );
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// returns -Infinity
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v = y0( 1.0 );
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// returns ~0.088
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v = y0( Infinity );
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// returns 0.0
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```
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If `x < 0` or `x` is `NaN`, the function returns `NaN`.
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```javascript
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var v = y0( -1.0 );
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// returns NaN
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v = y0( -Infinity );
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// returns NaN
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v = y0( NaN );
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// returns NaN
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var randu = require( '@stdlib/random/base/randu' );
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var y0 = require( '@stdlib/math/base/special/bessely0' );
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var x;
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var i;
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for ( i = 0; i < 100; i++ ) {
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x = randu() * 10.0;
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console.log( 'y0(%d) = %d', x, y0( x ) );
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}
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[bessel-second-kind]: https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_second_kind:_Y.CE.B1
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</section>
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<!-- /.links -->
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