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Why W?

Brian Hayes

Whence and Whither W?

The modern history of Lambert W began in the 1980s, when a version of the function was built into the Maple computer-algebra system and given the name W. Why W? An earlier publication by F. N. Fritsch, R. E. Shafer and W. P. Crowley of the Lawrence Livermore Laboratory had written the defining equation as wew = x. The Maple routine was written by Gaston H. Gonnet of the Institut für Wissenschaftliches Rechnen in Zurich, who adopted the letter w but because of typographic conventions in Maple had to capitalize it.

A few years later Robert M. Corless and David J. Jeffrey of the University of Western Ontario launched a discussion of W and its applications in what has turned out to be a long series of journal articles and less-formal publications. The most influential paper, issued as a preprint in 1993 but not published until 1996, was written by Corless and Jeffrey in collaboration with Gonnet, David E. G. Hare of the University of Waterloo and Donald E. Knuth of Stanford University. This was the paper that named the function in honor of the 18th-century savant Johann Heinrich Lambert.

Lambert, who wrote on everything from cartography to photometry to philosophy, never published a word on the function that now bears his name. It was his eminent colleague Leonhard Euler who first described a variant of the W function in a paper published in 1779, two years after Lambert's death. So why isn't it called the Euler W function? For one thing, Euler gave credit to Lambert for the earliest work on the subject. Perhaps more to the point, Corless, Jeffrey and Knuth note that "naming yet another function after Euler would not be useful."

In the years between Euler and Maple, the W function did not disappear entirely. The Dutch mathematician N. G. de Bruijn analyzed the equation in 1958, and the British mathematician E. M. Wright wrote on the subject at about the same time. In the 1970s and 80s there were several more contributions, including that of Fritsch, Shafer and Crowley. Nevertheless, the literature remained widely scattered and obscure until the function acquired a name. In a 1993 article, Corless, Gonnet, Hare and Jeffrey remark: "For a function, getting your own name is rather like Pinocchio getting to be a real boy."

Some of the recent publications on W go beyond mere explication of mathematics; they carry a whiff of evangelical fervor. Those for whom W is a favorite function want to see it elevated to the canon of standard textbook functions, alongside log and sine and square root. I am reminded of another kind of canonization—a campaign for the recognition of a local saint, with testimonials to good works and miracles performed.

The advocates of W do make a strong case. In a 2002 paper, Corless and Jeffrey argue that W is in some sense the smallest step beyond the present set of elementary functions. "The Lambert W function is the simplest example of the root of an exponential polynomial; and exponential polynomials are the next simplest class of functions after polynomials."

But the elevation of W has not won universal assent. R. William Gosper, Jr., has suggested that a better choice might be the square of W, that is, WeW2 = x, which eliminates the multivalued branching on the real line. (In a play on "Lambert W," Gosper calls this the Dilbert lambda function.) And Dan Kalman of American University has suggested a formulation based on eW/W = c, with an inverse function he calls glog.

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