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. If you’re instinct is that things with strange names are no longer important, you’re instinct might be backward. The strange name may be an indication that something is so important that its usefulness became apparent long ago.
Sometimes special functions have bland, uninformative names because the names stuck before anybody could think of something better. Bob looks into an interesting family of functions, then later he finds another interesting family of functions. These become known as “Bob’s functions of the first kind” and “Bob’s functions of the second kind.” These names are quite understandable at the time, though in the future people will want to know what distinguishes the functions, other than the fact that Bob discovered them, and what the groupings have in common other than the order in which Bob found them.
I started this post intending to discuss modified Bessel functions and explain what exactly is modified about them, but my preface became its own post. “Modified” is an example of the bland terminology mentioned above. There are Bessel functions and modified Bessel functions. Without more context, the “modified” term isn’t very informative. But it does provide a clue that there’s some kind of close relationship between the modified and unmodified functions. That’ll be the topic of my next post.