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Hermite polynomial

Contents

Contents

Idea

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Definition

For nn \in \mathbb{N} a natural number, the nnth Hermite polynomial is the polynomial in one variable xx with coefficients in the real numbers (and as such a naturally regarded as a smooth function \mathbb{R} \to \mathbb{R} on the real line) which is given by the following expression:

(1)H n(x)(1) nexp(12x 2)ddx nexp(12x 2), H_n(x) \;\coloneqq\; (-1)^n \exp\left( \tfrac{1}{2} x^2 \right) \frac{d}{d x^n} \exp\left( -\tfrac{1}{2} x^2 \right) \,,

where exp()\exp(-) denotes the exponential function and d/dx nd/d x^n denotes the nnth derivative with respect to xx.

Similarly, the Hermite functions are

(2)h n(x) (1) nn!2 nπexp(12x 2)ddx nexp(x 2) =1n!πH n(2x)exp(12x 2), \begin{aligned} h_n(x) & \coloneqq\; \frac{ (-1)^n }{ \sqrt{ n! 2^n \sqrt{\pi} } } \exp\left( \tfrac{1}{2} x^2 \right) \frac{d}{d x^n} \exp\left( - x^2 \right) \\ & = \frac{ 1 }{ \sqrt{ n! \sqrt{\pi} } } \, H_n \big( \sqrt{2} x \big) \, \exp \left( -\tfrac{1}{2}x^2 \right) \end{aligned} \,,

Properties

Orthonormality

Regarded as smooth functions h n:h_n \colon \mathbb{R} \to \mathbb{R} on the real line, the Hermite functions (2) form an orthonormal basis of square-integrable functions with respect to the 2-norm, in that the integration over their product satisfies

(3) 1h n(x)h m(x)dx=δ n,m \int_{\mathbb{R}^1} h_{n}(x) h_{m}(x) \, d x \;=\; \delta_{n,m}

(where on the right we have the Kronecker delta).

Recursion relations

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(4)ddxh n(x)=n2h n1(x)n12h n+1(x) \frac{d}{d x} h_n(x) \;=\; \sqrt{ \tfrac {n} {2} } h_{n-1}(x) - \sqrt{ \tfrac {n-1} {2} } h_{n+1}(x)

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References

Named after Charles Hermite.

  • Hermite polynomials and Hermite functions (pdf)

See also:

Last revised on May 4, 2020 at 12:57:50. See the history of this page for a list of all contributions to it.