As I teach partial differential equations and compose sections for a textbook on the subject, I often run across special families of functions that I rarely see in other contexts. Recently this involved the Hermite functions which are related to the Hermite polynomials. The linked notes assume the reader is familiar with the Hermite polynomials and points out some of the connections between the Hermite polynomials and the Hermite functions. To me the most amazing feature of the Hermite functions is that they are eigenfunctions of the Fourier Transform.
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