This is paper no. 4 in the list on p. . In 1992, David Jeffrey, Dave Hare, Gaston Gonnet and I wrote a paper on the function which satisfies , which we traced back to J. H. Lambert and named after him. The journal reviews of this paper were very complimentary, when we submitted it for publication, and it could have appeared already in 1994. Gaston Gonnet brought the paper to the attention of D. E. Knuth, who responded with a substantial collection of notes on this function which he had been working on himself for several years. It was very gratifying to find that we had been working on something that someone of Knuth's stature found interesting, and to have made progress that he admired.

We adopted some of Knuth's material into a revised version of the paper, and he became a co-author. This has since led to a fruitful collaboration, with two more papers with Knuth so far, and more in progress.

The material of the first **W** paper ranges in level from simple, *
i.e.* accessible
by even first-year undergraduates, to quite deep, and manages to show
most undergraduate mathematics in a new light along the way.
The significance of this work is perhaps not so much its depth but its
utility. The Lambert **W** function is very simple, and has applications
in an extraordinary number of fields, ranging from analysis of
algorithms in computer science through fluid mechanics to
pure mathematical results in dynamical systems.

Fri Jan 9 10:52:31 EST 1998