COMPUTING SCIENCE

# Why W?

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
*we ^{w}* =

*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, *We ^{W2}* =

*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

*e*=

^{W}/W*c*, with an inverse function he calls

*glog*.

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