7

According to this Wikipedia page, the Lambert W relation cannot be expressed in terms of elementary functions.

However, it does not explain why this is the case.

An elementary function is "a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots)," according to Wikipedia.

Note

I know that I should not always refer to the same source for information, but I believe that this is an accurate definition of the term.

Questions

Does there exist a proof that the Lambert W relation cannot be expressed in terms of elementary functions?

Why is it that the Lambert W relation cannot be expressed in terms of elementary functions?

J. M. ain't a mathematician
  • 75,051
Taylor
  • 1,128
  • The Lambert W function is a non-elementary function. It's called a non-elementary function because it's a non-elementary function. So... I think the answers you're going to get are largely going to be "because it's defined that way". – Zain Patel Jul 15 '15 at 21:48
  • 2
    @ZainPatel The OP is asking why there isn't some clever formula like $W(t) = \ln t/\ln\ln t$. – Stephen Montgomery-Smith Jul 15 '15 at 21:51
  • @StephenMontgomery-Smith In a way, yes, but, also, no, I am not. It is clear through working with the Lambert W relation that, as it does not yield any values, i.e. you cannot input a value, and have it return an exact value, unfortunately, but I am wondering why it is the case that it is impossible to derive a formula in order to do this. – Taylor Jul 15 '15 at 21:52
  • 1
    @Taylor But you have the same issue with the exponential and trig functions. – Stephen Montgomery-Smith Jul 15 '15 at 21:53
  • 1
    @Taylor : what do you means by "you cannot input a value, and have it return an exact value". If I understand what you mean, even with $x \mapsto \sqrt{x}$ "you cannot input a value, and have it return an exact value" – Tryss Jul 15 '15 at 21:54
  • @Tryss For example, let $f(x) = x^{2}.$ It is clear that, if you let $x = 3,$ and try to calculate $f(x),$ it is possible, and the answer is $9.$ However, say that you have the equation $e^{x}x = 3,$ and you want to find what $x$ is equal to. $x = W(3),$ but $W(y)$ cannot be calculated exactly. Of course, we have Newton's method, and Halley's method, but these are only approximations, albeit, if you take the approximation far enough, it will give a very good approximation. – Taylor Jul 15 '15 at 21:57
  • 2
    What's the difference between $\ln(3)$ and $W(3)$? Can you calculate exactly $\ln(3)$? – Tryss Jul 15 '15 at 21:58
  • @Taylor, But if you have the function $f(x) = \sqrt{x}$ and you let $x=2$, you're gonna get $\sqrt{2}$ which is not exact at all and you need approximations for it. – Zain Patel Jul 15 '15 at 21:59
  • @ZainPatel Yes, but is that not delving into the world of irrational numbers? – Taylor Jul 15 '15 at 22:00
  • The point I'm trying to make is that non-exact answers is not what defines a function to be non-elementary. – Zain Patel Jul 15 '15 at 22:00
  • @ZainPatel No, but it is the inability to obtain exact answers. – Taylor Jul 15 '15 at 22:01
  • 1
    Whaaaat? No. What's your definition for a function to be non-elementary? – Zain Patel Jul 15 '15 at 22:02
  • 5
    @Taylor, here you go: http://mathoverflow.net/questions/135911/how-to-prove-lamberts-w-function-is-not-elementary – Zain Patel Jul 15 '15 at 22:02
  • @ZainPatel A non-elementary function is a function of more than one variable which is the composition of an infinite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots). – Taylor Jul 15 '15 at 22:03
  • @Taylor : so nothing in this definition is about the "inability to obtain exact answers". – Tryss Jul 15 '15 at 22:04
  • @Tryss It is implied in that, if there are an infinite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations, it would be near impossible to find an exact answer. – Taylor Jul 15 '15 at 22:05
  • I believe the point is that when we talk about $\sqrt{x}$, the point is that it is an algebraic number over the rationals when we include $x$, so we call $\sqrt{x}$ an "elementary function." Similarly for natural log, exponential, and roots of algebraic equations, and any natural compositions of these using function composition or arithmetic operations. It's all about expressibility. The fact that the $W$ function is non-elementary is a statement about expressibility, not ability to approximate. I think OP deserves a proof or reference to a proof of this. – user2566092 Jul 15 '15 at 22:06
  • But most of the elementary functions don't gives "exact answers" as well. What's the "exact answer" of $x^5+4x^3-x^2+4 = 0$ – Tryss Jul 15 '15 at 22:07
  • @user2566092, I did provide a reference to the exact same question at mathoverflow: http://mathoverflow.net/questions/135911/how-to-prove-lamberts-w-function-is-not-elementary – Zain Patel Jul 15 '15 at 22:14
  • The article pointed to by Zain Patel costs $48 US. I'll trust the summary. – marty cohen Jul 15 '15 at 22:14
  • @martycohen How can people charge for a math paper? :'-( – Taylor Jul 15 '15 at 22:16
  • It's in a journal that charges for access. Go look at it. – marty cohen Jul 15 '15 at 22:17
  • @martycohen Ah, I did not mean literally, I meant morally. It is math, which is meant to be shared to all. :-) – Taylor Jul 15 '15 at 22:18
  • So this question is a duplicate of an MO question (as Zain Patel noted). But we cannot close the question unless there is a duplicate in this same forum. – GEdgar Jul 15 '15 at 22:29
  • But if this question is asked again, we can close it. – marty cohen Jul 15 '15 at 22:31
  • Maybe someone could make an answer copying the one on MO to have the relevant info not buried in the comments ;) – Tryss Jul 15 '15 at 22:34
  • @Taylor Lambert W provides us exact results but 99% of the times they are Transcendental numbers in the same ways logs do and many other elementary functions. Those results are exact and there are a lot of way to compute them in terms of elementary function (an infinite number of them) as Taylor expansion and Big theta representation. – AlienRem Jul 17 '15 at 16:50

2 Answers2

6

The branches of Lambert W are the local inverses of the Elementary function $f$ with $f(z)=ze^z$, $z \in \mathbb{C}$.

The incomprehensibly unfortunately hardly noticed theorem of Joseph Fels Ritt in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of Elementary functions can have an inverse which is an Elementary function. It is also proved in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759.
Ritt, J. F.: Integration in finite terms. Liouville's theory of elementary methods. Columbia University Press, New York, 1948

The non-elementarity of LambertW was already proved by Liouville in
Liouville, J.: Mémoire sur la classification des transcendantes et sur l'impossibilité d'exprimer les racines de certaines équations en fonction finie explicite des coefficients. Journal de mathématiques pures et appliquées 2 (1837) 56–105, 3 (1838) 5233–547
It is also proved in
Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22
and in
Bronstein, M.; Corless, R. M.; Davenport, J. H., Jeffrey, D. J.: Algebraic properties of the Lambert W Function from a result of Rosenlicht and of Liouville. Integral Transforms and Special Functions 19 (2008) (10) 709-712.

Ritt's theorem shows that no antiderivatives, no differentiation and no differential fields are needed for defining the Elementary functions.

IV_
  • 6,964
1

I would recommend that you could possibly look into hyper-operations. They are quite interesting and related. For quick reference, a hyper-operation of n is a repetition of they {n-1}th hyper-operation.$$5*4=5+5+5+5$$Here, multiplication was turned into a repetition of addition. Similarly, exponentiation is a repetition of multiplication. And so on...

But more importantly, all of this can be represented in terms of elementary functions. Whether it be addition, multiplication, or exponentiation.

Furthermore, the inverses of each individual hyper-operation is considered elementary. The opposites of addition, multiplication, and exponentiation are subtraction, division, and logarithms or roots, respectively.

However, a combination of different levels of hyper-operations (except addition) cannot be inverted. For example:$$f(x)=xe^x$$$$f^{-1}=?$$For this example, we assign the Lambert W function as the solution.$$f^{-1}=W(x)$$But the problem is that we cannot turn this into something involving only addition, multiplication, exponentiation, and their inverses (or higher hyper-operations like tetrations and such).

More specifically, we cannot turn this into a $finite$ amount of terms being added or multiplied and such.

To answer your first question, the proof is simply that the definition of the Lambert W function cannot be solved with elementary functions.

As to why, it is because, as far as I can explain, two hyper-operations, multiplication and exponentiation, were combined. In general, combining different hyper-operations results in unsolvable inverses (or a manipulation of the Lambert W function). Try solving (without the Lambert W function): $$f(x)=x^a+bx$$$$f^{-1}(x)=?$$Now try solving it with $a=3,2,1,0$. Much easier? To solve for any known $a$ is easy, but solving for all $a$'s is more difficult.

The reason why you were able to solve for the above values of $a$ were most likely because of factoring. However, you cannot factor with exponents like you can with polynomials. Envision the following:$$a^{(ax)^{(ax^2)^{..^{..^{..}}}}}$$Compare it to$$a+ax+ax^2+...$$$$and$$$$a*ax*ax^2*ax^3*...$$The last two are simplifiable, but the exponential one was not. This is why we stop being able to find inverses of functions when they are exponential, tetrational, or higher.

Simply Beautiful Art
  • 74,685