The heart of 19th century math was the study of special functions arising from mathematical physics.

It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations.

The above quote was the judgment ofÂ Felix Klein (of Klein bottle fame) in 1893. The differential equations he had in mind were the second order differential equations of mathematical physics.

Special functions were the core of 19th century math, and hypergeometric series were the grand unifying theory of special functions. (Not every special function is hypergeometric, but quite a few are.) And yet they’re hardly taught any more. I never heard of hypergeometric series in college, even though I studied differential equations and applied math. Later I encountered hypergeometric functions first in combinatorics and only later in differential equations.

It’s odd that what was “the central problem of the whole of modern mathematics” could become almost a lost art a century later. How could this be? I believe part of the explanation is that special functions, and hypergeometric function in particular, fall between two stools: too advanced for undergraduate programs but not a hot enough of a research area for graduate programs.

You’re right about this, of course. Guesses as to why this is so: 1) Much of modern math has been disassociated from physics. My graduate math classes included no references to physics that I can recall. 2) Some related topics in math are now considered old and musty. This is the “hotness” issue you point out. Theory of a single complex variable, for example. ODEs, to some extent. My grad school ODE text was Coddington and Levinson, which I think first appeared in 1955. 3) Computers. Both numeric and symbolic methods have helped alleviate the need for a strong knowledge of these functions, just as they have for good skills at computing definite intergals.

You may be interested in this speech by Wolfram on special functions: http://www.stephenwolfram.com/publications/recent/specialfunctions/

Thanks, Andrew. I ran across that speech recently and enjoyed reading it.

In my home college, in the first of two courses of Mathematical Methods for Physicists, I taught hypergeometric functions. That and Gamma functions are my favorites. If one use Mathematica, some results are in terms of hypergeometric series, so I thought that at least to teach it to have an idea of what it meant, understand it.

I feel like I’m two centuries back, either way I like it.

I’ve finally spent some time about a week or two ago to thoroughly learn hypergeometric functions and put my notes here:

http://theoretical-physics.net/dev/src/math/hyper.html

Today I discovered your blog, it looks like you went through the same thing a few years ago. I only learned a little bit about 2F1 in the QM course, but I though that’s just some not very useful thing. Only now I realized that it’s a nice unifying tool and it’s *simple*.

Yes! I ran across a very old Analysis textbook (Whittaker and Watson), that dealt extensively with hypergeometric functions, but trying to find anything recent on them is all but impossible.

Another one for the self study list once I’ve finished the MSc….

That’s true. I never even heard of the Alan Turing’s work or Hilbert’s Millennium problems in college.