COMPUTING SCIENCE

# Why W?

No, not *that* *W*. I won't be drawn into presidential
politics here. The *W* I want to discuss is something else
entirely: the Lambert *W* function, a mathematical
contrivance that has been getting a fair amount of attention lately.
The buzz began in the world of computer-algebra systems such as
Macsyma, Maple and Mathematica, but word of *W* has also been
spreading through journal articles, preprints, conference
presentations and Internet news groups. The *W* function even
has its own poster (see http://www.orcca.on.ca/LambertW).

The concept at the root of *W* can be traced back through
more than two centuries of the mathematical literature, but the
function itself has had a name only for the past 10 years or so. (A
few years longer if you count a name used within the Maple software
but otherwise unpublished.) When it comes to mathematical objects,
it turns out that names are more important than you might guess.

Without further ado, here is the definition of Lambert
*W*: It is the inverse function associated with the equation:

*We ^{W}* =

*x*.

What does that mean? Some readers of this column will grasp it
instantly, but I am not going to pretend that I am one of them. It
took me a while to figure out how *W* works, and even longer
to see why the concept might be considered interesting or important.
At the risk of inflicting severe tedium on those who are more adept
at algebra and analysis, I want to retrace my own path toward
understanding what *W* is all about. It's a fairly long and
wiggly path.

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