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 Incoming links: Charles Hermite

Charles Hermite Created 2026-10-20 Updated 2026-10-24

A mathematician whose name is attached to Hermite polynomials.

Figure 1.
Portrait of Charles Hermite
. Photograph by Eugène Pirou (1841–1909); heliogravure by Paul Dujardin (1843–1913); printer Ch. Whittmann; published in Correspondance d’Hermite et de Stieltjes, 1905. Public domain: origin and life-plus-70 jurisdictions; US pre-1931 publication. Source usage terms, via Wikimedia Commons. Source.

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Hermite polynomial Created 2026-09-24 Updated 2026-11-05

The physicists' Hermite polynomials are

H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d ^{n}}{d x ^{n}} e^{- x^{2}} .

They are orthogonal on the real line with weight

e^{- x^{2}}

. The first examples are

1

,

2 x

and

4 x^{2} - 2

; they solve the Hermite differential equation

H_{n}^{''} - 2 x H_{n}^{'} + 2 n H_{n} = 0

.

Multiplication by a Gaussian factor converts them into quantum harmonic-oscillator eigenfunctions. The probabilists' normalization instead has weight

e^{- x^{2} /2}

and a rescaled variable. Stating the convention prevents factors of two from being hidden in the name. The polynomials are named after Charles Hermite.^[ref]

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