Hermite function

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A solution of the Hermite equation

The Hermite functions have the form

where is the contour in the complex -plane consisting of the rays and and the semi-circle , , and . The half-sum of these solutions,

for an integer , is equal to the Hermite polynomial (cf. Hermite polynomials). The name Hermite equation is also used for

When is an integer, this equation has the fundamental system of solutions , where are the Hermite polynomials and are the Hermite functions of the second kind, which can be expressed in terms of the confluent hypergeometric function:


[1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 1 , Interscience (1953) (Translated from German)
[2] A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)


The Hermite functions and are related to the parabolic cylinder functions (cf. Parabolic cylinder function). See [a1], Sect. 4b for some further results concerning the functions when is a non-negative integer.


[a1] L. Durand, "Nicholson-type integrals for products of Gegenbauer functions and related topics" R.A. Askey (ed.) , Theory and Application of Special Functions , Acad. Press (1975) pp. 353–374
How to Cite This Entry:
Hermite function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article