Why special function terminology is arcane

Special functions are special because they’re useful. They can also be shrouded in arcane terminology. These two facts are related. The more widely useful a function is, the more likely it is that the function will be discovered independently multiple times. Independent discoveries lead to varying definitions and notations. For example, there are two widely used definitions of Hermite polynomials, one used in probability and another used in physics, that only differ by a scaling factor. This also explains why there are so many variations on the definitions of the Fourier transform and spherical coordinates. Special functions were discovered and applied before they were studied systematically. As with most mathematics, practice preceded theory. In hindsight, some names and conventions were less than ideal, at least from the perspective of someone seeking to organize a theory. Functions can have arcane names for several reasons, one being that their usefulness became apparent long ago.…