<!--

@license Apache-2.0

Copyright (c) 2018 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.

-->

# y0

> Compute the [Bessel function of the second kind][bessel-second-kind] of order zero.

<section class="intro">

The [Bessel function of the second kind][bessel-second-kind] of order zero is defined as

<!-- <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"> -->

<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">
    <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">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### y0( x )

Computes the [Bessel function of the second kind][bessel-second-kind] of order zero at `x`.

```javascript
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`.

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

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

v = y0( NaN );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var randu = require( '@stdlib/random/base/randu' );
var y0 = require( '@stdlib/math/base/special/bessely0' );

var x;
var i;

for ( i = 0; i < 100; i++ ) {
    x = randu() * 10.0;
    console.log( 'y0(%d) = %d', x, y0( x ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

[bessel-second-kind]: https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_second_kind:_Y.CE.B1

</section>

<!-- /.links -->