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# Physicist's Hermite Polynomial

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

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<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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<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>

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<section class="notes">

</section>

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<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>

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</section>

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* * *

## 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>

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<section class="links">

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

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[@stdlib/math/base/tools/evalpoly]: https://github.com/stdlib-js/math/tree/main/base/tools/evalpoly

[@stdlib/math/base/tools/normhermitepoly]: https://github.com/stdlib-js/math/tree/main/base/tools/normhermitepoly

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</section>

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