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COMPUTING SCIENCE

Why W?

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 a solution.

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.