145 lines
3.6 KiB
Markdown
145 lines
3.6 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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# Physicist's Hermite Polynomial
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> Evaluate a physicist's [Hermite polynomial][hermite-polynomial].
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<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
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<section class="intro">
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The physicist's [Hermite polynomials][hermite-polynomial] are given by
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<!-- <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."> -->
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<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">
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<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.">
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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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<!-- Package usage documentation. -->
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<section class="usage">
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## Usage
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```javascript
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var hermitepoly = require( '@stdlib/math/base/tools/hermitepoly' );
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```
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#### hermitepoly( n, x )
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Evaluates a physicist's [Hermite polynomial][hermite-polynomial] of degree `n`.
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```javascript
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var v = hermitepoly( 1, 1.0 );
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// returns 2.0
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v = hermitepoly( 1, 0.5 );
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// returns ~1.0
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v = hermitepoly( -1, 0.5 );
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// returns NaN
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v = hermitepoly( 0, 0.5 );
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// returns 1.0
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v = hermitepoly( 2, 0.5 );
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// returns -1.0
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```
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#### hermitepoly.factory( n )
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Returns a `function` for evaluating a physicist's [Hermite polynomial][hermite-polynomial] of degree `n`.
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```javascript
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var polyval = hermitepoly.factory( 2 );
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var v = polyval( 0.5 );
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// returns -1.0
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```
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</section>
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<!-- /.usage -->
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<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="notes">
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</section>
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<!-- /.notes -->
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<!-- Package usage examples. -->
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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 hermitepoly = require( '@stdlib/math/base/tools/hermitepoly' );
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var x;
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var y;
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var i;
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var j;
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for ( i = 0; i < 100; i++ ) {
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x = (randu()*100.0) - 50.0;
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for ( j = 1; j < 3; j++ ) {
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y = hermitepoly( j, x );
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console.log( 'H_%d( %d ) = %d', j, x.toFixed( 3 ), y.toFixed( 3 ) );
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}
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}
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```
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</section>
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<!-- /.examples -->
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<!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="references">
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</section>
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<!-- /.references -->
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<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="links">
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[hermite-polynomial]: https://en.wikipedia.org/wiki/Hermite_polynomials
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</section>
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<!-- /.links -->
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