npm package: @stdlib/math-base-tools-hermitepoly
Description: Evaluate a physicist's Hermite polynomial.
Author: The Stdlib Authors
Latest version: 0.2.3

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# Physicist's Hermite Polynomial

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> Evaluate a physicist's [Hermite polynomial][hermite-polynomial].

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

The physicist's [Hermite polynomials][hermite-polynomial] are given by

<!-- <equation class="equation" label="eq:physicists_hermite_polynomials" align="center" raw="H_{n}(x)=(-1)^{n} e^{x^2} \frac{\mathrm{d}^{n}}{\mathrm{d} x^{n}} e^{-x^2}" alt="Equation for physicist's Hermite polynomials."> -->

<div class="equation" align="center" data-raw-text="H_{n}(x)=(-1)^{n} e^{x^2} \frac{\mathrm{d}^{n}}{\mathrm{d}x^n} e^{-x^2}" data-equation="eq:physicists_hermite_polynomials">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@58b02120eb58818177f6767ab495e7afac3618e8/lib/node_modules/@stdlib/math/base/tools/hermitepoly/docs/img/equation_physicists_hermite_polynomials.svg" alt="Equation for physicist's Hermite polynomials.">
    <br>
</div>

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</section>

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<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/math-base-tools-hermitepoly
```

</section>

<section class="usage">

## Usage

```javascript
var hermitepoly = require( '@stdlib/math-base-tools-hermitepoly' );
```

#### hermitepoly( n, x )

Evaluates a physicist's [Hermite polynomial][hermite-polynomial] of degree `n`.

```javascript
var v = hermitepoly( 1, 1.0 );
// returns 2.0

v = hermitepoly( 1, 0.5 );
// returns ~1.0

v = hermitepoly( -1, 0.5 );
// returns NaN

v = hermitepoly( 0, 0.5 );
// returns 1.0

v = hermitepoly( 2, 0.5 );
// returns -1.0
```

#### hermitepoly.factory( n )

Returns a `function` for evaluating a physicist's [Hermite polynomial][hermite-polynomial] of degree `n`.

```javascript
var polyval = hermitepoly.factory( 2 );

var v = polyval( 0.5 );
// returns -1.0
```

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random-base-randu');
var hermitepoly = require( '@stdlib/math-base-tools-hermitepoly' );

var x;
var y;
var i;
var j;

for ( i = 0; i < 100; i++ ) {
    x = (randu()*100.0) - 50.0;
    for ( j = 1; j < 3; j++ ) {
        y = hermitepoly( j, x );
        console.log( 'H_%d( %d ) = %d', j, x.toFixed( 3 ), y.toFixed( 3 ) );
    }
}
```

</section>

<!-- /.examples -->

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<section class="references">

</section>

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<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math-base/tools/evalpoly`][@stdlib/math/base/tools/evalpoly]</span><span class="delimiter">: </span><span class="description">evaluate a polynomial using double-precision floating-point arithmetic.</span>
-   <span class="package-name">[`@stdlib/math-base/tools/normhermitepoly`][@stdlib/math/base/tools/normhermitepoly]</span><span class="delimiter">: </span><span class="description">evaluate a normalized Hermite polynomial using double-precision floating-point arithmetic.</span>

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-tools-hermitepoly/main/LICENSE

[hermite-polynomial]: https://en.wikipedia.org/wiki/Hermite_polynomials

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[@stdlib/math/base/tools/evalpoly]: https://www.npmjs.com/package/@stdlib/math-base-tools-evalpoly

[@stdlib/math/base/tools/normhermitepoly]: https://www.npmjs.com/package/@stdlib/math-base-tools-normhermitepoly

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