1. Introduction
The Lambert W function, named after Johann Heinrich Lambert [1], is a standard function in both Mathematica, where it’s called Product
, and in Maple, where you can use both Lambert
or Lambert
. The zero in this latter expression denotes the principal branch of the inverse of
. The actual usage of the letter W has a rather vague origin. One source attributes it to some earlier papers on the subject that wrote the standard equation as
using a small w. Programming protocol with Maple then forced the letter to be capitalized [2]. Another source [3] attributes the W to honor the British mathematician Sir Edward M. Wright (famous co-author with G. H. Hardy of An Introduction to the Theory of Numbers) who did a lot of pioneering work with the function. Finally, Robert Corless and David Jeffrey of the University of Western Ontario have written, during the past several decades, a number of journal articles on the function. Their paper in 1996, in collaboration with Gaston Gonnet, David Hare, and Donald Knuth, was where Lambert’s name got attached to the function [2]. It could have been coined the Euler W function, since Euler had studied the equation
[4] (although Euler credits Lambert as studying the equation first [5]), but they decided Euler had enough items attached to his name!
2. Definition
The exponential function
is defined for all real x, but has a codomain of
. This function (Figure 1(a)) is the product of two elementary functions, each defined on the entire real line, and each being one-to-one; but the product is not injective. Consequently, if we restrict the domain to
, then
will possess an inverse, which is a function, and it’s this function that is now known as the (principal) Lambert W function (Figure 1(b)), written as
. An alternative branch for W would be defined for that portion of
when
. We won’t consider that situation in this article.
Several function values of W are easy to computesuch as
.
and
. The value of
, known as the omega constant, has the approximate value 0.567143. The number
is, in some sense, a distant cousin of the golden ratio
, since
is a solution to
, and
is the solution to
, and
is the linear Maclaurin approximation to
(Figure 2). Since W is the inverse of
, it follows that
and that the slope of the curve in Figure 1(b) at the point
is
.
3. Computation
A natural question is how to compute arbitrary values of
. One result, from the Lagrange inversion theorem, asserts that the Lambert W function has the Taylor series expansion [6,7]
(1)
which, unfortunately, has a radius of convergence of merely
. Since the denominator n! grows rapidly it’s
advantageous to write the series with the coefficients defined recursively as
, with
. This recursion lends itself to easy programming evaluation. Testing this, with say a series of 150 terms (which is plenty, considering that
), with
, we obtain a partial sum value of
, which differs from the exact value of
by 0.0000003. We also note that
, so the use of the series is justified.
On the other hand, a TI-graphing calculator returns “overflow error” if we try to determine
, primarily since the coefficients grow rapidly.
Suppose that
and we wish to compute
.
One possibility is the series
(2)
where
and
denotes a Stirling number of the first kind [3]. The series (2) is somewhat impractical to use because of the difficulty in determining
; it turns out to be more useful to employ some standard numerical schemes for approximating
.
First, setting
, we need to solve
. Defining the function g by
, we use Newton’s method to approximate y in
. This gives
. To determine
, for example, starting with an initial approximate of
, after 7 more iterations we get
, which is an excellent approximation to
because
returns 2 on the calculator. If x is a relatively small number, then an initial approximate of 0 will suffice for the algorithm; but if x is large, then ln x can be chosen for
. For instance, if
, choose
, and after 5 iterations we get W(10) ≈ 1.745528003.
Newton’s method is a favorite iteration scheme for many because of its simplicity, though the convergence, quadratic in general, is typically relatively slow. A faster choice is furnished by Halley’s method (of Halley’s comet fame), which produces cubic convergence, and happens to be the choice implemented by the software Maple; this scheme gives [8]
![](https://www.scirp.org/html/5-7401300\533ed231-3943-496e-b1e7-b6bdf8af51a1.jpg)
Employing this gives W(10) ≈ 1.745528003 after 3 iterations. This complex looking scheme is actually what you get when you apply Newton’s method to the function
[9]. An alternative root-finding scheme, using continued fraction expansion, is described in [10].
4. Calculus
We know that since
is an increasing and differentiable function for all
then its inverse
is likewise increasing and differentiable for all
.
Differentiating this latter equation with respect to y, we obtain
![](https://www.scirp.org/html/5-7401300\7c0ffc74-e3f5-460c-8455-5fe38caab9e3.jpg)
so
(3)
In particular,
, and similarly,
. What about
the right-hand side of (3) is indeterminant at
, but division of both sides of (1) by x and taking the limit as
give
. This yields
![](https://www.scirp.org/html/5-7401300\81d4652b-d7f3-4e69-8317-20cc97739866.jpg)
For large x, the graph of
bears strong resemblance to
, since from (2) we have
although we have to be careful here because the difference
increases without bound as
[7]. The graph of
, like that of
, is concave downward for all x since
is concave upward. If we differentiate (3), and omit the argument x for brevity, then
![](https://www.scirp.org/html/5-7401300\a22cdaba-a85f-4fc0-8cd0-508fc98a8f45.jpg)
Rewriting
as
puts this into the form which fits the general case for
[5]. In fact, from this form, we readily see that there is a point of inflection on the curve when
, which actually falls on the other branch of the W function.
Continuing along the calculus vein, we should examine, if possible, the integral of
. To this end, recall that
iff
. Thus,
![](https://www.scirp.org/html/5-7401300\70e3ade5-05ad-4801-a427-b8ad00da03e8.jpg)
and integrating this last integral by parts, we obtain
, which now gives
(4)
In particular, the area of the region bounded by the curve
, the x-axis, and the line
is, therefore,
![](https://www.scirp.org/html/5-7401300\5adf1bd4-0f6d-4e89-93be-aefe76560f09.jpg)
We note this result agrees with evaluating the integral via inverse functions [11], because then
![](https://www.scirp.org/html/5-7401300\fe709c7b-858b-4591-8b99-ed5ad915060a.jpg)
Other integrals, involving functions containing W, can be computed, some just with a special change of variable [6]. For instance,
.
The function
is concave up, connecting ![](https://www.scirp.org/html/5-7401300\2eefc331-b3df-472b-b10f-dcee3f2e55df.jpg)
and
, hence its area
is less than
. Similarly we find
, and this is greater than
since
is increasing and concave down from
to
.
5. Applications
An article appeared in the February, 2000, issue of FOCUS, the newsletter of the Mathematical Association of America, touting the merits of the W function as a candidate for a new elementary function to be studied in schools and to be included in textbooks [12]. The rationale for this was that not only is W a radically different function from the traditional elementary functions of polynomials, rationals, exponentials, logarithmics, and trigonometrics, but its calculus provides a wealth of interesting, and powerful, applications. A number of these are mentioned in a paper by Corless et al., where they describe such applications as enumeration of trees, combustion, enzyme kinetics, linear delay equations, population growth, spread of disease, and the analysis of algorithms [3]. An article [13] by Packel and Yuen shows that W is instrumental in determining the maximum range for a projectile with linear resistance (problems of this type have certainly been important for several thousand years). The solution for the current in a series diode/resistor circuit can also be written in terms of W. Applications of W are found in complex cases involving atomic, nuclear, and optical physics. The first physics problem to be solved explicitly in terms of W was one in which the exchange forces between two nuclei within the hydrogen molecular ion
were calculated [14]. Several other cases involve generalized Gaussian noise, solar winds, black holes, general relativity, quantum chromodynamics, fuel consumption, Stirling’s formula for n!, cardiorespiratory control, water-wave heights in oceanography, enumeration of trees in combinatorics, and statistical mechanics [5,15-17]. A really interesting analog of
is given by Dan Kalman [18], where he defines a function glog, similar to W, in that glog is the inverse to
. The glog function bears a strong resemblance to W, possessing similar properties and useful common applications, such as solving exponential-linear equations. The two functions are intimately related by
and
.
In the remainder of this article I wish to focus on a couple of applications dealing with ordinary algebraic equation solving.
6. Algebra
In a high-school precalculus course one might be presented with the elementary equation
to solve. Now, instead, let’s solve a similar equation
, which means that it won’t suffice to begin by taking the logarithm of both sides. Instead, we proceed as follows:
![](https://www.scirp.org/html/5-7401300\2542f47d-02b5-4e44-a756-31b42824ea4f.jpg)
Since the right-hand side of this last equation is of the form
, and since we know
iff
then
, or
.
Using Kalman’s glog function we can solve ![](https://www.scirp.org/html/5-7401300\0a85dbe2-d9a4-45ca-935c-e969584b8c53.jpg)
and get
. Since
we can use (1) to approximate
and get −0.0746900848, so x = 0.1077550149. Checking, we find 2x = 1.07755015 = 10x.
The equation
is a special case of a more general setting
, where we assume the base
and where neither b nor d equals zero. The substitution
then gives
![](https://www.scirp.org/html/5-7401300\e85ce785-c9cf-4655-a790-51c3b485ae79.jpg)
and, thus,
![](https://www.scirp.org/html/5-7401300\d6034735-73e0-4f96-a751-a662df0c8f15.jpg)
Multiplication of both sides by
gives
, which now has the form
, so
or ![](https://www.scirp.org/html/5-7401300\3eefb194-14c9-45af-8379-20f05056557e.jpg)
and, hence,
![](https://www.scirp.org/html/5-7401300\b24d4fae-23fc-40f8-b839-a5dbf9d38f4e.jpg)
that is, ![](https://www.scirp.org/html/5-7401300\dcd418e7-4c10-423d-908f-73502c83dfef.jpg)
Another interesting algebraic application involves the infinite tower of exponents
, which will be denoted by
. To solve the particular equation
one might argue that this is equivalent to
, in which case we have
, so
, which is the correct solution to
But what about T(x) = 3, T(x) = 4, or T(x) = y. It stands to reason that as y increases, so does x. But with
, we can write this as
, or
, so
again! Something isn’t right.
The problem lies with the domain of T. We find in [19] that the infinite tower of exponents is only defined (i.e.its interval of convergence) for
, or approximately
. So if x is selected from this interval, what is
? If we set
then
.
Note also that
when
, and the above expression for y gives a function continuous at
since
. Hence, if
, then
, so
, and this is why the equation
is solvable, but
is not.
The graph of T is therefore an increasing function with domain
and range
. It also passes through the two obvious points of
and
. What else can we deduce? Checking for differentiability, we have from (3),
![](https://www.scirp.org/html/5-7401300\c5d9d20f-a772-4a88-ae32-ae874a0aef89.jpg)
and since the limit of this expression is 1 as
, then
, and hence
is never 0, so T is always strictly increasing. The following small table (Table 1) of values will prove helpful.
Alternatively, we could have found
by implicit differentiation of
. Thus
or
; so again
. Use of this form for easier access to
then gives, after some algebraic manipulations and cancellations,
![](https://www.scirp.org/html/5-7401300\0d4c2d1d-3647-4e01-a7f9-6a502e4026e7.jpg)
This complex expression appears to yield negative values for all
and positive values for all
, and
. Hence, we have an inflection point at
. Also
. Putting all of these pieces of the puzzle together, we obtain a decent graph of T, as shown in Figure 3.
The tower function T must necessarily possess an inverse
. We note then that
![](https://www.scirp.org/html/5-7401300\1ad5995e-4885-45cc-bd66-375a3fea6f17.jpg)
and, consequently, this inverse is
. Composition of the two functions give the interesting pair of identities,
![](https://www.scirp.org/html/5-7401300\6efeb4d9-862c-4193-bdee-c50ed3ced4bd.jpg)