Difference between revisions of "Hermite (physicist)"

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The (physicist) Hermite polynomials are a sequence of [[orthogonal polynomials]] with weight function $e^{-x^2}$.  
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The (physicist) Hermite polynomials $H_n$ are a sequence of [[orthogonal polynomials]] with weight function $w(x)=e^{-x^2}$ over $(-\infty,\infty)$.
  
 
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$$\begin{array}{ll}
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H_0(x)=1\\
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H_1(x)=2x\\
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H_2(x)=4x^2-2\\
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H_3(x)=8x^3-12x\\
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H_4(x)=16x^4-48x^2+12\\
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H_5(x)=32x^5-160x^3+120x
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\end{array}$$
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=Properties=
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[[Generating function for Hermite (physicist) polynomials]]<br />
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[[Closed formula for physicist's Hermite polynomials]]<br />
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[[Hermite (physicist) polynomial at negative argument]]<br />
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[[Recurrence relation for physicist's Hermite involving derivative of H_n, derivative of H_n-1, and H_n]]
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<strong>Theorem:</strong> (Orthogonality) Let $w(x)=e^{-x^2}$, then
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$$\displaystyle\int_{-\infty}^{\infty} H_n(x)H_m(x)w(x) dx=\sqrt{\pi}2^nn!\delta_{mn},$$
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where $H_n$ denotes the [[Hermite (physicist)|Hermite polynomials]] and $\delta_{mn}$ denotes the [[Kronecker delta]].
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<strong>Proof:</strong> █
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<strong>Theorem:</strong> The following formula holds:
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$$H_{n+1}(x)=2xH_n(x)-H_n'(x).$$
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<strong>Proof:</strong> █
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<strong>Theorem:</strong> The following formula holds:
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$$H_n'(x)=2nH_{n-1}(x).$$
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<strong>Proof:</strong> █
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<strong>Theorem:</strong> The following formula holds:
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$$H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x).$$
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<strong>Proof:</strong> █
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</div>
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</div>
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{{:Orthogonal polynomials footer}}
 
{{:Orthogonal polynomials footer}}
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[[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 $(-\infty,\infty)$.

$$\begin{array}{ll} H_0(x)=1\\ H_1(x)=2x\\ H_2(x)=4x^2-2\\ H_3(x)=8x^3-12x\\ H_4(x)=16x^4-48x^2+12\\ H_5(x)=32x^5-160x^3+120x \end{array}$$

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:


Orthogonal polynomials