What is so special about special functions?
Recently, a student asked me, “Why is the gamma function called a special function?” I shot back, “Because it has some important properties that make it special!”. Being an obedient student, he accepted this vague answer and thanked me. But it got me thinking.
I listed a few of the so called special functions
- Gamma function \(\Gamma(x)\),
- Bessel functions \(J_n(x)\) and \(Y_n(x)\),
- Legendre polynomials \(P_n(x)\),
- Hermite polynomials \(H_n(x)\),
- Laguerre polynomials \(L_n(x)\), etc.
So they are not special because they are non-polynomial functions.
By the way, non-polynomial functions are called transcendental functions. So these special functions are transcendental functions. But there are many other transcendental functions that are not classified as special functions, for example, the exponential function \(e^x\) and the logarithm function \(\log(x)\).
I also noticed that these have some common properties:
- They are orthogonal functions.
- They satisfy some differential equations.
- They have some recurrence relations.
- They have some integral representations.
- They have some series representations.
- They have some asymptotic expansions.
Then the next obvious question is: are trigonometric functions special functions?
Lets see.
They are orthogonal functions \(\int_0^{2\pi} \sin(nx) \sin(mx) dx = 0\) for \(n \neq m\) and \(\int_0^{2\pi} \cos(nx) \cos(mx) dx = 0\) for \(n \neq m\).
They frequently appear in the solution of differential equations like the wave equation, the heat equation, etc.
They have some recurrence relations like \(\sin((n+1)x) = 2\cos(x)\sin(nx) - \sin((n-1)x)\) and \(\cos((n+1)x) = 2\cos(x)\cos(nx) - \cos((n-1)x)\).
They have some integral representations like \(\sin(x) = \frac{1}{2i} \int_{-\infty}^{\infty} e^{itx} dt\) and \(\cos(x) = \frac{1}{2} \int_{-\infty}^{\infty} e^{itx} dt\).
They have some series representations like \(\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\) and \(\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\).
They have some asymptotic expansions like \(\sin(x) \sim \frac{1}{2i} e^{ix}\) and \(\cos(x) \sim \frac{1}{2} e^{ix}\) as \(x \to \infty\).
So they should be classified as special functions. But you will see that they are not classified as special functions.
So clearly something fishy is going on.
As far as I have gathered from discussions with my colleagues, and of course from the internet, the term “special functions” is a historical term that was coined in the 19th century to refer to a class of functions that were considered important and useful in various branches of mathematics and physics. The term “special” does not necessarily mean that these functions are more special than other functions, but rather that they have some special properties that make them useful in certain contexts. By this time, people were already familiar with trigonometric functions and exponential functions, and they were not considered as special functions. The term “special functions” was used to refer to a new class of functions that were being studied at that time, and it has stuck ever since.
On a related note, the trigonometric functions were first studied by the ancient Greeks, while the exponential and logarithm functions were first studied by the ancient Babylonians and the ancient Egyptians.
So next time you hear the term “special functions”, don’t be alarmed. It is an historical artifact that has been passed down from generation to generation of mathematicians and physicists. It is not a technical term that has a precise definition, but rather a convenient label for a class of functions that have some special properties.