Expansions in terms of Parabolic Cylinder Functions
- 1 October 1948
- journal article
- research article
- Published by Cambridge University Press (CUP) in Proceedings of the Edinburgh Mathematical Society
- Vol. 8 (2) , 50-65
- https://doi.org/10.1017/s0013091500024792
Abstract
When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the form where x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.Keywords
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