# Festschrift for Oleg Marichev

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.