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:

References

 [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)