W—What Is It Good For?
The Lambert W function may make a pretty curve, but what's
it good for? Why should anyone care? By mixing up a few symbols we
could generate an endless variety of function definitions. What
makes this one stand out from all the rest?
If you ask the same question of more familiar functions such as
exp and log and square root, the answer is that
those functions are tools useful in solving broad classes of
mathematical problems. With just the four basic operations of
arithmetic, you can represent the solution of any linear equation.
Adding square roots to the toolbox allows you to solve quadratic
equations as well. Expanding the kit to include the trigonometric,
exponential and logarithmic functions brings still more problems
within reach. All of these well-known functions, and perhaps a few
more, are classified as "elementary." The exact membership
of this category is not written in stone, but it excludes more
specialized tools such as Bessel functions.
A few years ago, a brief, unsigned editorial in Focus, the
newsletter of the Mathematical Association of America, asked:
"Time for a new elementary function?" The function
proposed for promotion to the core set was Lambert W.
Whether W ultimately attains such canonical status will
depend on whether the mathematical community at large finds it
sufficiently useful, which won't be clear for some years. In the
meantime, I can list a few applications of W discovered so far.
One place where W turns up in pure mathematics
is the "power tower," the infinitely iterated exponential
For large x, this expression soars off to infinity faster
than we can follow it, but Euler showed that the tower converges to
a finite value in the domain between x =
e-e (about 0.07) and x =
e1/e (about 1.44). Within this realm, the value
to which the infinite tower converges is W(-log(x))/-log(x).
W has another cameo role in the "omega constant,"
which is a distant of cousin of the golden ratio. The latter
constant, with a value of about 1.618, is a solution of the
quadratic equation 1/x= x-1. The omega constant is
the solution of an exponential variant of this equation, to wit:
1/ex = x. And what is the value of that
solution? It is W(1), equal to about 0.567143.
Of more practical import, W also appears in solutions to a
large family of equations known as delay differential equations,
which describe situations where the present rate of change in some
quantity depends on the value of the quantity at an earlier moment.
Behavior of this kind can be found in population dynamics, in
economics, in control theory and even in the bathroom shower, where
the temperature of the water now depends on the setting of the
mixing valve a few moments ago. Many delay differential equations
can be solved in terms of W; in some cases the two branches
of the W function correspond to distinct physical solutions.
A recent article by Edward W. Packel and David S. Yuen of Lake
Forest College applies the W function to the classical
problem of describing the motion of a ballistic projectile in the
presence of air resistance. In a vacuum, as Galileo knew, the
ballistic path is a parabola, and the maximum range is attained when
the projectile is launched at an angle of 45 degrees. Air resistance
warps the symmetry of the curve and greatly complicates its
mathematical description. Packel and Yuen show that the projectile's
range can be given in terms of a W function, although the
expression is still forbiddingly complex. (They remark:
"Honesty compels us to admit at this point that the idea for
using Lambert W to find a closed-form solution was really
Mathematica's and not ours.")
Still another example comes from electrical engineering, where T. C.
Banwell of Telcordia Technologies and A. Jayakumar of Anadigics show
that a W function describes the relation between voltage,
current and resistance in a diode. In a simple resistor, this
relation is given by Ohm's law, I=V/R, where I is
the current, V the voltage and R the resistance.
In a diode, however, the relation is nonlinear: Although current
still depends on voltage and resistance, the resistance in turn
depends on current and voltage. Banwell and Jayakumar note that no
explicit formula for the diode current can be constructed from the
elementary functions, but adding W to the repertory allows
Other applications of W have been discovered in statistical
mechanics, quantum chemistry, combinatorics, enzyme kinetics, the
physiology of vision, the engineering of thin films, hydrology and
the analysis of algorithms.