The Lambert W-Function

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In a recent post ( The Power Tower ) we described a function defined by iterated exponentiation:

$\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$

It would seem that when ${x>1}$ this must blow up. Surprisingly, it has finite values for a range of x>1.

Below, we show that the power tower function may be expressed in terms of a function called the Lambert W-Function.

The W-function has applications in a wide range of areas in pure and applied mathematics. Thus, in addition to being a source of innocent merriment, the power tower function is connected with many important practical problems.

Johann Heinrich Lambert

Johann Heinrich Lambert (1728–1777) was a Swiss mathematician, physicist and astronomer. Lambert was born about twenty years later than Euler. In one of his papers, Euler referred to his younger compatriot as “The ingenious engineer Lambert”.

Johann Heinrich Lambert (1728–1777).

Lambert is remembered as the first person to prove the irrationality of ${\pi}$ . Euler had earlier proved that e is irrational. Lambert conjectured that e and ${\pi}$ were both transcendental numbers. But the proof of this was not found for about another century. The transcendence of ${e}$ was shown in 1873 by Charles Hermite and, in 1882, Ferdinand von Lindemann published a proof that ${\pi}$ is transcendental.

Lambert had very wide scientific interests. He introduced the hyperbolic functions into spherical geometry and proved some key results for hyperbolic triangles. He also devised several map projections that are still in use today.

Lambert conformal conic projection with standard parallels at 20 N and 50 N (image from Wikimedia Commons).

The Lambert W-Function

In studying the solutions of a family of algebraic equations, Lambert introduced a power series related to a function that has proved to be of wide value and importance. The Lambert W-function is defined as the inverse of the function ${z=w\exp(w)}$ . Thus

$\displaystyle w = W(z) \qquad\Longleftrightarrow\qquad z = w\exp(w) \,.$

A plot of ${w=W(z)}$ is presented here.

Lambert W-function w=W(z), defined as the inverse of z=w exp(w).

We confine attention to real values of ${W(z)}$ , which means that ${z\ge -1/e}$ . The W-function is single-valued for ${z\ge 0}$ and double-valued for ${-1/e<z<0}$ . The constraint ${W(z)>-1}$ defines a single-valued function on ${z\in[-1/e,+\infty)}$ . This is the principal branch, denoted when appropriate as ${W_0(z)}$ (shown in blue above). The other branch, real on ${z\in[-1/e,0)}$ , is denoted ${W_{-1}(z)}$ (shown in red).

Applications of the W-Function

The Lambert W-function occurs frequently in mathematics and physics. Indeed, it has been “re-discovered’ several times in various contexts. In pure mathematics, the W-function is valuable in solving transcendental and differential equations, in combinatorics (as the Tree function), for delay differential equations and for iterated epxonentials (which is the context in which we have introduced it).

In theoretical computer science, the W-function is used in the analysis of algorithms. Physical applications include water waves, combustion problems, population growth, eigenstates of the hydrogen molecule and, recently, quantum gravity.

The W-function also serves as a pedagogical aid. It is a useful example in introducing implicit functions. It is also a valuable test case for numerical solution methods. In the context of complex variable theory, it is a simple example of a function with both algebraic and logarithmic singularities.

Finally, W(z) has a range of interesting asymptotic behaviours. For further references on this, see the technical note linked at the end of this post.

The Power Tower Function and W

For the power tower function defined at the top of this post, we consider the iterative sequence of successive approximations:

$\displaystyle y_1 = x \, \qquad y_{n+1} = x^{y_n}.$

Through numerical experiments, we find that the sequence ${\{y_n\}}$ converges for ${e^{-e}<x<e^{1/e}}$ . In fact, this result was first proved by Euler! When this sequence converges, we have an explicit expression for ${x}$ as a function of ${y}$ :

$\displaystyle x = y^{1/y}$

Defining ${\xi=\log x}$ , it follows that ${y=\exp(\xi y)}$ . We can write this as

$\displaystyle (-\xi y)\exp(-\xi y) = (-\xi)$

We now define ${z = -\xi}$ and ${w = -\xi y}$ and have ${z=w\exp(w)}$ . But, by the definition of the Lambert W-function, this means that ${w=W(z)}$ .

Returning to variables ${x}$ and ${y}$ , we conclude that

$\displaystyle y = \frac{W(-\log x)}{-\log x}$

which is the expression for the Power Tower function in terms of the Lambert W-function.

Further Information

Technical Note with more detail:The Power Tower Function. Peter Lynch, UCD, 2013 (PDF).

The canonical reference on the Lambert W-Function: Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996). On the Lambert W function. Adv. Comp. Math. 5, 329–359 (Preprint (Postscript) )
A lighthearted introduction to the Lambert W-function: Hayes, Brian, 2005: Why W? American Scientist, 93, 104-108 ( PDF ).

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