Why Special Functions are called 'special'?

Question

Why Special Functions are called 'special' ?

What particular thing made it so special ?

Answer 1

Himself once asked this question. For myself, I described the term "special functions" as follows: such that arise from "uncountable" integrals, solving transcendental and differential equations (as an example, you can specify the solution of the system of equations characteristic modes for the volt-watt characteristic of solar cells), expressing the properties of numbers ( functions with factorials) and functions with unusual properties. Their irreducibility to a combination of elementary functions sometimes makes it impossible to operate with such functions in an analytical form, and to carry out only numerical calculations. I hope my version of the definition of special functions will help you.

Answer 2

"Special" is to be seen as being opposed to "general", in the sense of "select".

"General functions" refers to arbitrary functions - in the case of real number functions defined on $\mathbb{R}$ or a subset thereof, arbitrary functions comprise an enormous set with size $\beth_2$ (equiv. the size of the power set of the real numbers, or the number of subsets of real numbers, or the number of binary "strings" with continually many bits).

"Special functions" are a few select functions from that set which have some sort of "interesting" properties for some or another reason and given certain names or standard symbols. It's a pretty open-ended term, and new ones are coined all the time.

Answer 3

The basic operations $+,-,\times,\div$ and exponentiation in the complex numbers give rise to all familiar transcendental functions

$$e^x,\cosh x,\sinh x, \tanh x,\cos x,\sin x,\tan x,\sec x,\csc x,\cot x.$$

And by functional inversion,

$$\log x,\text{arcosh }x,\text{arsinh }x,\text{artanh }x,\arccos x,\arcsin x,\arctan x$$ and a few others.

They form the set of functions usually allowed in closed-form expressions. They can be called elementary.

All other distinguished functions, such as Gamma, Beta, Bessel, Elliptic, Error, Fresnel, Zeta... (more on NIST) are called special. This is a catch-all category. More subcategories here.