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Why W?

Brian Hayes

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

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 (A few years longer if you count a name used within the Maple software but otherwise unpublished.)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:

WeW = 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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