# What's the deal with Lambert W function questions?

I have seen a large amount of questions that are:

solve $x^2=2^x$

Why are there so many of these questions? Who's assigning these questions? Why did I never see them when I took high school algebra? Are they just for personal interest? If so, why are so many of them so convoluted looking?

I have never seen the Lambert W function in my experiences in undergrad/now grad school. I see it fairly routinely on MSE. Why is the Lambert W function so overrepresented on MSE?

• It's not overrepresented, here. It's not even underrepresented elsewhere, simply folks there wouldn't understand it.
– user436658
Nov 20, 2017 at 19:43
• I share this sentiment. Lambert W was totally absent during my undergrad education as well as grad school (the latter is hardly a surprise). The non-elementary functions we learned about were the likes of Bessel functions, Gamma, Erf and such that were actually needed in various places (physics, number theory, whatnot). I guess times have changed??? Nov 20, 2017 at 19:45
• I can't say it was totally missing from my undergraduate studies. But I think that it was only brought up once in either measure theory or functional analysis.
– Asaf Karagila Mod
Nov 20, 2017 at 20:37
• I would not imagine anyone is assigning such things. We get an enormous slopover from other websites, several kids see something on a website and post it here, paraphrasing in random ways. Nov 20, 2017 at 20:43
• My guess is that some people wrongly copy the problem by either omitting details like $x$ is a natural number or they are asked how much solutions are there instead of actually needing to solve the equation. Nov 20, 2017 at 20:47
• What @kingW3 said; or, they didn't notice that they were being asked to solve numerically. Some of it may just be curiosity. You sketch the graph of $y=e^x$, you sketch the graph of $y=-x$, you see they intersect somewhere, you wonder just where that somewhere is. Or, you try to find a local maximum for some innocent-looking function by setting the derivative to zero, and you get an equation custom-made for Lambert W. Nov 20, 2017 at 22:51
• I don't find the Lambert function very interesting as opposed to other non-elementary functions like Gamma, Zeta, polylog, elliptic/theta/hypergeometric functions. When I see an equation like the one in your question, I don't think beyond techniques for numerical solution (like @GerryMyerson said above). I wonder if Lambert function has any interesting connections with other more popular functions. Questions involving these equations do not appear to be well thought out.
– Paramanand Singh Mod
Nov 21, 2017 at 5:55
• The first time I saw Lambert W was on Math.SE. Since then it's just popped up all over the place. Nov 21, 2017 at 6:01
• I'm a combinatorist and although I know that some combinatorists use it for some stuff about tree counting, I think it came up once or twice across all undergraduate or graduate courses I've taken. It has come up a bunch in musings I've had as a high schooler, but nothing for coursework or research. I graduated in 2016, so I don't think this is a time thing. Nov 21, 2017 at 17:04
• I think certain topics attract more attention in online discussion groups than in formal classes, and this has been true since at least the early 1990s (on sci.math). Other examples are tetration, proofs that require the Axiom of Choice, non-computable functions such as the Busy Beaver function, fractals (self-similar kinds), deconstructing (and often just "disproving") the Cantor diagonal argument for uncountability of the reals, etc. If anything, in the last 7 or 8 years with math stack exchange and mathoverflow, online math discussions have gotten MUCH closer to standard textbook topics. Nov 21, 2017 at 19:36
• Now is about the time when Newton's Method is being taught in Calc 1. Possibly they are popping up in that context, or in the context of the MVT being used to demonstrate the existence of unique solutions. Nov 22, 2017 at 18:49
• The natural symmetry of the variable and constants in $x^2 = 2^x$ seems a likely cause for curiosity. As I recall when I first had access to a computer algebra system it was among the first non-trivial equations I asked it to solve, expecting an answer in terms of known functions, only to discover the lambert W function. This might play a part in its rise in popularity post-internet. Nov 25, 2017 at 13:10
• I don't know what questions you have in mind and I've been inactive from MSE, so what I'm about to say may very well not apply. I remember the first time I wanted to solve such an equation. I had no idea what the W-Lambert function was (nor would I understand it at the time), I just wanted to prove that $x\mapsto x^2-2^x$ had a zero somewhere and it didn't occur to me to use Bolzano's theorem. So this sort of question might not be a goal, but a means to an end. Nov 27, 2017 at 1:09
• To second the comment about combinatorics, W shows up when studying Bell and Stirling numbers and (set) partitions; for instance, it turns out to be relevant to calculating (approximate) means and modes of certain distributions associated with them. See volume 4 of The Art of Computer Programming for some details. Nov 29, 2017 at 19:54
• The published version of @DuchampGérardH.E.'s article: Corless et al. - On the Lambert $W$-function. (Note that 'et al.' hides Knuth.) Nov 29, 2017 at 21:01

I don't know, it seems to me a very natural question to arise, not from homework, but from a student's own curiosity, while first learning algebra. Maybe they tried plotting the functions on a graphing calculator and see that there is an answer, but are stuck trying to find it themselves with the tools they know. Maybe they asked their teacher and were told, "it's impossible," giving the problem an air of mystery that also drives interest in dividing by zero, formulas for roots of high-degree polynomials, summation methods for divergent series, etc.

The other answers show, that there is a certain universality to human curiosity (especially the instinct to generalize). When you know just enough high school math to start playing around with it, there seem to be a very limited number of questions that people ask themselves. When you start learning about powers, especially when $x$ starts going into the exponent, you go and investigate the difference between $a^x$ and $x^a$ (or even make a few mistakes this way).

I myself noticed a classmate making mistake between $x^2$ and $2^x$ and immediately asked myself when he'd actually be right. I noticed there is a third root in addition to $2$ and $4$ (got there by plotting), and was fascinated to find out it's "impossible" (along the way, discovered the power-tower solution and found the intersection manually with bisection on graph paper). Generalization to $a^x=x^a$ was even more interesting (discovering that $a=e$ special case has a double solution in positives - the magic $e$ strikes again). I even went so far to try to solve this in excel (high-school, didn't know the tools), and made a t-shirt with the negative solution $-0.76666469596212309311...$ printed on it.

Don't forget teenagers tend to think big and exaggerate. Dealing with $a^x$ and $x^a$ (and differences in how you differentiate them), you try being clever by going "what about $x^x$? Which rule applies now?". This of course again quickly leads to Lambert's function.

I'd say, these are the "popular" interesting math questions high-schoolers reinvent over and over again independently:

• Reciting $\pi$ (for some reason, always $\pi$, almost never $e$ or some other constant - there's just something attractive about it).
• $x^x$, $x=e^x$ and variations (Lambert function type)
• $x=\cos x$, $ax=\sin x$ and $x=\tan x$ (trig-type transcendentals, shocked to find out it can't be analytically solved)
• Generalized mean $((x^n+y^n)/2)^{1/n}$, maybe leading to the limiting case, which gives you the geometric mean. Alternatively, generalized Pythagorean theorem (no no division involved).
• Infinite sums (usually related to Riemann Zeta, geometric or similar "obvious" sequences, sometimes leading to $1+2+3+\cdots$), again, shocked to see it's not straight forward to compute.

When I was young, I amused myself with various mathematical problems I came up with on my own, including finding numerical solutions to interesting equations such as $x^x = 2$. At that time, I would just perform binary search to find the solution to the precision on my calculator. It was of interest to me to find out that the solution could be expressed in terms of the Lambert W function, despite not appearing to be in the same form.

As for your specific equation in your question, if $x$ is a positive real, it can be proven elementarily that $x^2 = 2^x$ iff $x = 2$ or $x = 4$. In particular, all you need is to show that the function $f : \mathbb{R}^+ \to \mathbb{R}$ defined as $f(x) = x^{1/x}$ for real $x > 0$ has a unique maximum at $e$. So sometimes the question may have a simple solution that has nothing to do with the Lambert W function.

• There's also a negative solution $\approx-0.7666647$, so your "iff" is wrong. Dec 4, 2017 at 10:09
• @Ruslan: Argh; you are right. I was thinking "positive real" but wrote only "real". I've edited my post, thanks! Dec 4, 2017 at 10:40

I think these questions became popular with the rise of the calculator. When we first started using Maple and Mathematica in the classroom, we needed problems that still required thinking. (I was against the tendency to create complicated problems just so Maple wouldn't instantly solve them and force the student to think the problem through. It seemed an easier, much cheaper solution to just remove the CAS.) One way was to use unfamiliar functions. This became a never-ending cycle, because the instant we found something Maple didn't know, the people at Waterloo would add it.

If you want a student to understand how graphs and solutions are related, then these are pretty good problems for that. You can test understanding of the inverse function theorem and some Taylor series things, too, with Lambert-type problems.

The most common way to solve equations of one variable in closed form is to guess particular solutions or to apply the inverse operations that can be read from the equation.

The latter method is applicable if the function of the equation is a composition of functions of one variable with known inverse functions. Algebraic operations are algebraic functions.

The elementary functions are generated from their arguments by applying finite numbers of $$\exp$$, $$\ln$$ and/or unary or multiary $$\mathbb{C}$$-algebraic functions.

If we have an equation of the form $$A(f_1(x),...,f_n(x))=y,\tag{1}$$ where $$n>1$$, $$A$$ an algebraic function of $$n$$ variables, $$y$$ a constant or a variable, and $$f_1,...,f_n$$ are elementary functions of one variable that are algebraically independent, we don't know how to solve the equation by rearranging for $$x$$ by applying only elementary functions we can read from the equation.

It can be proven that the function on the left-hand side of the equation $$A(x,e^x)=y\tag{2}$$ cannot have partial inverses that are elementary functions.

Moreover, for functions $$A$$ for which equation (2) leads to an equation $$P(x,e^x)=0\$$ (3), where $$P$$ is an irreducible polynomial function of two complex variables with algebraic coefficients, it was proven that equation (3) doesn't have nonzero solutions $$x$$ that are elementary numbers.

That means, equations of the form (2) cannot be solved by applying elementary functions, except by $$x=0$$.

But such kinds of equations occur more frequently. Lambert W ist the most simple, most described and most known representative of inverse relations for equations of the type of equation (1).

"The survey paper of Corless et al.  describes a large number of applications of W. This function appears in the combinatorial enumeration of trees, in the jet fuel problem, in enzyme kinetics, or in the solution of delay differential equations, just to mention a few areas. Some specific applications of W in the investigation of solar wind comes from Cranmer , some applications in electromagnetic behavior of materials are given by Houari , other appearances in electromagnetics are investigated by Jenn . It is known that Wien’s displacement constant can be expressed by W, and in the discussion of capacitor fields W also appears . The Lambert function has applications in quantum statistics, too . A simple mathematical application connects W to the distribution of primes via the Prime Number Theorem ."

[Mező/Baricz 2017]:
"In 2002 R. M. Corless and D. J. Jeffrey wrote that 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". Thus, says B. Hayes , W is in some sense the smallest step beyond the present set of elementary functions". Based on these thoughts, it can happen that the complexity" of the root of an exponential polynomial strongly depends on the degree of the polynomial, as is the case for the classical polynomials. Speculatively, a new class of functions would arise in which W is just the simplest" member (in the Corless-Jeffrey sense) ..."

[Vazquez-Leal et al. 2020]:
"Later on, J.H. Lambert, in 1758, provided the principles for a new transcendental function (𝑊(𝑥)) that was almost unknown until 1990s, including its applications to science and engineering. This function was fully developed until 1925 by Pólya and Szegö  moreover, thanks to the algorithms implemented by Corless & developers , this function began to be widely employed by the end of 20th century."
$$\$$

How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?

[Mező/Keady 2015] Some physical applications of generalized Lambert functions. 2015

[Mező/Baricz 2017] Mező, I.; Baricz, Á.: On the generalization of the Lambert W function. Trans. Amer. Math. Soc. 369 (2017) 7917-7934

[Vazquez-Leal et al. 2020] Vazquez-Leal, H.; Sandoval-Hernandez, M. A.; Filobello-Ninoa, U.: The novel family of transcendental Leal-functions with applications to science and engineering. Heliyon 6 (2020) (11) e05418

• It seems this answer is about applications of the function instead of describing why there is a multitude of questions about it on math SE or why it was not introduced in courses. Nov 23 at 1:40
• @ТymaGaidash Oh yes. The frequency of questions about Lambert W is demonstrated i.a. by its importance for its applications. My answer answers all questions of the questioner except the question you mentioned.
– IV_
Nov 23 at 15:35