Why You Should Care about the Obscure
Mathematica has always had the most complete collection of special functions available. You might think that by now there were no more to add, but the next release of Mathematica will add another five. You might also think that any that are left to add are too obscure for you to care about. They are getting fairly obscure, but you should still care.
Let’s look at one of them: Owen’s T function.
This function was created by Donald Owen Bruce, an American statistician who stood out during the Cold War because of his knowledge of Russian statistical advances, which at the time would not have been considered good for his career. Owen’s career spanned the entire Cold War period, from his entry into the American Mathematical Society in 1948 (as cited in the Bulletin of the American Mathematical Society meeting minutes) through his last book in 1988, Beating Your Competition through Quality.
The Owen T function dates from his theoretical work in the 1950s, before he moved on to applications in oil exploration and early contributions to manufacturing quality processes, which would eventually evolve into techniques such as six-sigma.
Wikipedia manages only a single formula. Not much recognition for a function that has been around for over 50 years.
But special functions are not added to Mathematica for their own sake, or for some collector’s desire for completeness; they are added because they are useful. The T function is involved in all kinds of calculations related to probability and statistics. Because Mathematica automates computations, you might not know when it is being used, but the accuracy, speed, and even the solvability of your tasks may depend on it. Such as this probability question:
Sometimes it might even surprise you by popping up in your answers:
One of the reasons for its obscurity has been the difficulty in working with it; there are a handful of machine-precision FORTRAN libraries for numeric calculations, but any more than that and you need to start programming. Mathematica will change that. Every function in Mathematica gets a full implementation (carefully integrated with every part of the system)…
…as well as exact values…
…and symbolic rules:
Perhaps when it is just another part of the computational infrastructure, the function will become more popular, in the same way that you routinely see ProductLog in models in a way that might have seemed very exotic 20 years ago.