Difference between revisions of "Hermite (physicist)"
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| − | The (physicist) Hermite polynomials are a sequence of [[orthogonal polynomials]] with weight function $e^{-x^2}$. | + | The (physicist) Hermite polynomials $H_n$ are a sequence of [[orthogonal polynomials]] with weight function $w(x)=e^{-x^2}$ over $(-\infty,\infty)$. |
| − | + | ||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Hermiteh(physicist).png|Graph of $H_n$ on $[-4,4]$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | $$\begin{array}{ll} | ||
| + | H_0(x)=1\\ | ||
| + | H_1(x)=2x\\ | ||
| + | H_2(x)=4x^2-2\\ | ||
| + | H_3(x)=8x^3-12x\\ | ||
| + | H_4(x)=16x^4-48x^2+12\\ | ||
| + | H_5(x)=32x^5-160x^3+120x | ||
| + | \end{array}$$ | ||
| + | |||
| + | =Properties= | ||
| + | [[Generating function for Hermite (physicist) polynomials]]<br /> | ||
| + | [[Closed formula for physicist's Hermite polynomials]]<br /> | ||
| + | [[Hermite (physicist) polynomial at negative argument]]<br /> | ||
| + | [[Recurrence relation for physicist's Hermite involving derivative of H_n, derivative of H_n-1, and H_n]] | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> (Orthogonality) Let $w(x)=e^{-x^2}$, then | ||
| + | $$\displaystyle\int_{-\infty}^{\infty} H_n(x)H_m(x)w(x) dx=\sqrt{\pi}2^nn!\delta_{mn},$$ | ||
| + | where $H_n$ denotes the [[Hermite (physicist)|Hermite polynomials]] and $\delta_{mn}$ denotes the [[Kronecker delta]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$H_{n+1}(x)=2xH_n(x)-H_n'(x).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$H_n'(x)=2nH_{n-1}(x).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
{{:Orthogonal polynomials footer}} | {{:Orthogonal polynomials footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 00:49, 9 July 2016
The (physicist) Hermite polynomials $H_n$ are a sequence of orthogonal polynomials with weight function $w(x)=e^{-x^2}$ over $(-\infty,\infty)$.
Graph of $H_n$ on $[-4,4]$.
.png)
$$\begin{array}{ll} H_0(x)=1\\ H_1(x)=2x\\ H_2(x)=4x^2-2\\ H_3(x)=8x^3-12x\\ H_4(x)=16x^4-48x^2+12\\ H_5(x)=32x^5-160x^3+120x \end{array}$$
Properties
Generating function for Hermite (physicist) polynomials
Closed formula for physicist's Hermite polynomials
Hermite (physicist) polynomial at negative argument
Recurrence relation for physicist's Hermite involving derivative of H_n, derivative of H_n-1, and H_n
Theorem: (Orthogonality) Let $w(x)=e^{-x^2}$, then $$\displaystyle\int_{-\infty}^{\infty} H_n(x)H_m(x)w(x) dx=\sqrt{\pi}2^nn!\delta_{mn},$$ where $H_n$ denotes the Hermite polynomials and $\delta_{mn}$ denotes the Kronecker delta.
Proof: █
Theorem: The following formula holds: $$H_{n+1}(x)=2xH_n(x)-H_n'(x).$$
Proof: █
Theorem: The following formula holds: $$H_n'(x)=2nH_{n-1}(x).$$
Proof: █
Theorem: The following formula holds: $$H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x).$$
Proof: █
