113 lines
2.7 KiB
Markdown
113 lines
2.7 KiB
Markdown
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<!--
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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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# y1
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> Compute the [Bessel function of the second kind][bessel-second-kind] of order one.
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<section class="intro">
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The [Bessel function of the second kind][bessel-second-kind] of order one is defined as
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<!-- <equation class="equation" label="eq:bessel_second_kind_order_one" align="center" raw="Y_1(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - \theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left[ e^t - e^{-t} \right] e^{-x \sinh t} \, dt" alt="Bessel function of the second kind of order one"> -->
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<div class="equation" align="center" data-raw-text="Y_1(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - \theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left[ e^t - e^{-t} \right] e^{-x \sinh t} \, dt" data-equation="eq:bessel_second_kind_order_one">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/bessely1/docs/img/equation_bessel_second_kind_order_one.svg" alt="Bessel function of the second kind of order one">
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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 y1 = require( '@stdlib/math/base/special/bessely1' );
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```
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#### y1( x )
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Computes the [Bessel function of the second kind][bessel-second-kind] of order one at `x`.
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```javascript
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var v = y1( 0.0 );
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// returns -Infinity
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v = y1( 1.0 );
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// returns ~-0.781
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v = y1( 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 = y1( -1.0 );
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// returns NaN
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v = y1( -Infinity );
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// returns NaN
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v = y1( 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 y1 = require( '@stdlib/math/base/special/bessely1' );
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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( 'y1(%d) = %d', x, y1( 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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