View source for Hermite function

A solution of the [[Hermite equation|Hermite equation]]

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The Hermite functions have the form

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469804.png" /> is the contour in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469805.png" />-plane consisting of the rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469807.png" /> and the semi-circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469809.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698010.png" />. The half-sum of these solutions,

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for an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698012.png" />, is equal to the Hermite polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698013.png" /> (cf. [[Hermite polynomials|Hermite polynomials]]). The name Hermite equation is also used for

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When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698015.png" /> is an integer, this equation has the [[Fundamental system of solutions|fundamental system of solutions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698017.png" /> are the Hermite polynomials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698018.png" /> are the Hermite functions of the second kind, which can be expressed in terms of the [[Confluent hypergeometric function|confluent hypergeometric function]]:

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''1''' , Interscience  (1953)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Krazer,   W. Franz,   "Transzendente Funktionen" , Akademie Verlag  (1960)</TD></TR></table>



====Comments====
The Hermite functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698022.png" /> are related to the parabolic cylinder functions (cf. [[Parabolic cylinder function|Parabolic cylinder function]]). See [[#References|[a1]]], Sect. 4b for some further results concerning the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698023.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698024.png" /> is a non-negative integer.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR></table>