TITLE: What are special functions for?
QUESTION [20 upvotes]: If you read enough mathematics, you eventually come across several so-called "special functions". I'm always left wondering what on Earth these things are actually for.
We have the Euler Gamma function, the Beta function, the infamous Riemann zeta function, the Airy functions, the Möbius function, the Bessel functions, the Jacobi theta functions, the digamma and polygamma functions, the polylog, the Chebyshev polynomials, Weierstrass's elliptic functions, the Dedekind eta function, the hypergeometric functions, etc etc etc.
If you try looking any of these functions up, you generally get a huge page of dense, complex-looking formulas recounting the sixteen different ways to define the function, the myriad of properties and identities it has, and so on and so forth. But nowhere does it mention why this is useful.
I have just about figured out that the Euler Gamma function extends the factorial to non-integer values. I still don't know how that's "useful" in some way. Bessel J somehow appears at random in a few unrelated contexts, as do the Chebyshev polynomials. And that's about as far as I've got.
Obviously, I'm not expecting anyone to give an exhaustive explanation of what every single special function ever invented actually does. I suppose that if you don't know what a function does, that means you probably don't need it yet. But I would like to have some basic intuition as to why all these millions of functions have been defined, and what people use them for.

REPLY [3 votes]: There is a paper for general public published on SCRIBD : "Safari in the Contry of Special Functions", pp.18-36 : http://www.scribd.com/JJacquelin/documents