The History and Future of Special Functions

(Wolfram Technology Conference 2005, Champaign IL)

I’d like to welcome you to the first-ever Wolfram Research Festschrift. For those of you who don’t know, Festschrifts are a rather charming academic tradition. That definitely predate modern HR practices. They’re kind of intellectual birthday parties. That celebrate the work of people who’ve reached milestone ages.

Well, we thought it’d be nice to invent a corporate version. And I’m sure there are going to be a lot of these at our company in the future. Because in the 18 years that our company has been around, I’m happy to say that we’ve accumulated a large number of leaders in many fields.

But the first one to have reached a ripe enough age for a Festscrift is Oleg Marichev. So we’re here today to celebrate the work of Oleg Marichev.

And my own way of doing that is going to be to talk about Oleg’s favorite topic: special functions. But before starting on that, I wanted to say a little about Oleg himself. There’s a detailed bio that tells the whole personal story, interwoven with world events, that you can pick up.

I think I first heard about Oleg in 1985. When someone told me that there was now a table of integrals larger than Gradsteyn-Rhyzik. And that a man called Marichev was behind it. I’d long been an enthusiast of integrals. But at the time, I was probably in the longest integral-free period in my life, working on things like cellular automata.

But, in 1986 I started working on Mathematica. And in 1988 version 1 came out. And so I started caring about integrals again.

Then in early 1990 I get a letter about Oleg Marichev. And after a peculiar exchanges of faxes with the Minsk Watch Factory and so on, we arrange for Oleg and his student Victor Adamchik to come and visit us.

When they first arrived I think they didn’t know quite what to make of an American high-tech company. I remember them asking questions like whether the “Inc.” at the end of the name Wolfram Research, Inc. meant that Wolfram Research was incorporated into a larger company, like IBM.

And I remember them talking about Meijer G functions. And telling us—with a certain degree of mystery—that all those integrals in the big books of tables could be done just using Meijer G functions.

It sounded kinda crazy. I mean, I’d heard of Meijer G functions. But barely. And now I was being told that to do essentially any integral, one just had to write the integrand in terms of Meijer G functions. And then there’d essentially just be a formula for what the integral was.

There were algorithms known—and we’d implemented them—for all sorts of indefinite integrals of elementary functions. But for special functions and definite integrals, it was mostly just a bag of tricks. Nothing systematic.

So I thought this seemed kinda interesting. So I said “Sure, let’s see if we can make this Meijer G thing work in Mathematica“. So Oleg and Victor joined our company, and set about making Mathematica do integrals using Meijer G’s.

At the time—soon after the fall of the Berlin wall—we had a lot of talented former Soviet scientists and mathematicians working at our company. And some of the locals were particularly taken with it when a second Oleg joined our then-quite-small company.

There were definitely some Soviet features to whole thing, though. Like I remember seeing Oleg Marichev in the early days hard at work … banging away on a computer keyboard like it was a sticky Soviet typewriter. I think those keyboards lasted about a month each. Some kind of product of hard work, hard typing, and keyboard resilience.

But after a while the Meijer-G-based code for integration began to take shape. And within a few years, for example, we were able to do all the indefinite integrals in Gradsteyn-Rhyzik. No computer had ever come close to that before. And we even published errata to the Gradsteyn-Rhyzik tables, that Oleg’s wife Anna helped with.

And we kept going, building more and more capabilities for integration. Discovering lots of new ideas and algorithms. As I’ll talk about later, there used to be a whole culture of people who did difficult integrals by hand. But in the course of the nineties, I think they all got converted to Mathematica.

So the world’s whole difficult-integrals industry is now going through our Meijer G technology.

It’s kind of funny to think of all the kids who use the Integrator in calculus season—that it’s all fancy Meijer G hypergeometric-type functions inside.

Well, when you’re in the integrals business, you’re inevitably also in the special functions business. I’ll be talking a lot more about special functions in a few minutes.

But we’ve always taken special functions very seriously in Mathematica. And so, for example, in 1990 we made this poster about the zeta function.

We did that poster for the International Congress of Mathematicians, which is a fancy math conference that happens every 4 years.

Then in 1994 we did Solving the Quintic with Mathematica.

Well, for 1998 I thought we’d do something I’d long thought about: a poster about the 250 or so special functions in Mathematica, and the relations between them.

So Michael Trott and Oleg Marichev started working on it. At first, it was going to be a modest poster. But gradually it started growing. More relations were getting discovered by the day. And pretty soon it wasn’t just one poster, but five.

Then the type size went down. To 6. To 4.

With 9000 formulas altogether. Well, we started making preliminary versions. They stretched all the way down a long corridor. But they needed to be still bigger—27 feet long altogether. Here are some of them…..

And here’s a picture, including some of the designers and editors who worked on it..

Well, we started making a study of math-related organizations around the world, and how much wall space they had…. And the set of possible printers who had large enough presses was getting narrower and narrower. And eventually we realized that this really was an insane project.

In the lore of our company, it’s often viewed as the most out-of-control project we’ve ever had. And it’s made us scared about certain kinds of projects—and particularly posters—ever since.

But fortunately there’s the web. And from the poster was born our Functions site.

Which now has 87,000 relations about special functions on it. With a great many more ready to go in soon.

It’s become by far the world’s largest repository of mathematical function information. Mostly, I should say, created directly with Mathematica.

It’s really a very impressive project. As I’ll talk about in a few minutes, there’s a long and distinguished history of tables of formulas about mathematical functions. That represents the fruits of an immense number of mathematical careers. But there’s never been anything even close to our Functions site.

It’s a tribute to the tools we’ve built and the people who’ve worked on it. It’s a remarkable resource that I think we’re going to be able to get ever more from in the years to come. And of course it’s great that we have a company that lets us work on projects as long term as this.

You know, just last week I asked what was going on with it.”Oh, right now we’re working on the parametric derivatives of the inverse Jacobi functions”, I was told. Every little fact has human effort behind it. But every fact, once found, will last forever.

There was at one time quite a tradition of Soviet integral table makers. Oleg may be the last of them.

He’s been a prolific producer of mathematical formulas. In the past, in his books.

But for 14 years now, in Mathematica. Where—as we found out with the poster project—the productivity of formulas per day is so much higher. We’re happy to have had the benefit of so much of Oleg’s mathematical output. And we wish him many happy returns.