Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of "the functions one finds on a calculator"?

The function

$f(x)=x+\sin(x)$

is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map.

Now $g$ will almost certainly be a function which is not expressible using "the functions in a high-schooler's toolkit" (by which I guess I mean $\exp$, $\log$, and, if you like, the usual trigonometric functions and their friends like $\sinh$, although of course these can all be of course built from exponentials anyway). For purely recreational reasons (stemming from conversations I've had whilst teaching undergraduates) I'm interested in how one proves this sort of thing.

A few years ago I was interested in a related question, and took the trouble to learn some differential Galois theory. My motivation at the time was learning how to prove things like why $h(t):=\int_0^t e^{x^2} dx$ is not expressible in terms of these calculator-button functions (I'm sure there's a better name for them but I'm afraid I don't know it). I've realised that since then I've forgotten most of what I knew, but furthermore I am also unclear about whether this is the way one is supposed to proceed. Is the idea that I come up with some linear differential equation satisfied by $g$ and then apply some differential Galois theory technique? In fact, one of the many things that I have forgotten is the following: if $F$ is a field equipped with a differential operator $D$, and $E/F$ is the field extension obtained by adding a non-zero root of $Dh=ch$, with $c\in F$, then the Galois group of $E/F$ is solvable, whereas the equation itself might not be, in terms of calculator-button functions, if I can't integrate $c$.

Can a more enlightened soul explain to me how one is supposed to proceed? I wonder whether I am somehow conflating two ideas and the differential Galois theory business is a red herring, but it seemed simpler to ask rather than continuing to flounder around.

Solution 1:

This paper, titled "Elementary functions and their inverses" by J.F. Ritt addresses your question.

Some time ago, in searching for why some functions don't have elementary integrals, I was led to the work of Liouville, as digested by Ritt in his book "Integration in finite terms; Liouville's theory of elementary methods". It was written in 1948 so I think the copyright has expired, and you can find a download link via a Google search.

Liouville's results on elementary integrals were derived using quite basic tools (it is hard to be precise here on what I mean by "basic tools", best you see for yourself). The latter portions of Ritt's book explore elementary solutions of differential equations, which is based on the work of mathematicians after Liouville, and it is only from then that some differential Galois theory is used.

Ritt's paper uses methods somewhat similar to Liouville and in particular, does not seem to use differential Galois theory. However, it is possible there may be a more modern approach to your question that does use differential Galois theory, since the generalization of Liouville's original work develops it.

Alternatively, if you can express your inverse function in terms of the Lambert W function as Nicholas suggests, then you can answer your question via the more specific methods of this paper.

Solution 2:

To answer this, let's express $\sin(x)$ as $\frac i 2(e^{-ix}-e^{ix})$ so we now have $x+\frac i2(e^{-ix}-e^{ix})$. And you said that you are trying to find the inverse. From what I can see, this is relatable to $x+e^x = y$, the inverse of which is $x-W(e^x)$ where $W(x)$ is the product log function or Lambert $W$ function which cannot be expressed in terms of quote "the functions one finds on a calulator". Unfortunately, I am only in high school and therefore my answer may not be correct and I apologize if it is. This is only what I have gathered from my knowledge.

Solution 3:

I have finally found an answer to my own question, so I'll answer it myself and make the answer community wiki so I don't profit from it.

Let me first explain why the other answers given here did not completely resolve the problem. Let me use standard notation -- an elementary function is a funtion you can build using the buttons on your calculator. If you allow complex coefficients then basically you only need exp and log, as you can build everything else from this.

Ragib Zaman's answer points us to a paper of Ritt where he proves a theorem of the form "an elementary function whose inverse is also an elementary function must be of a very special form", but this very special form is quite hard to work with: it is basically of the form exp(rational function(log(rational function(log(rational function(exp(...))))))) where at each stage you can choose whether to use exp or log. The problem with this is that it's very hard (for me) to prove that the function $x+\sin(x)$ can provably not be expressed in this form.

Ritt's work relies on ideas of Liouville, and Liouville had a lot of ideas about this sort of question. Liouville proved a criterion for when an elementary function had an elementary integral (in fact he proved several results of this form, best phrased nowadays in the language of differential fields). One can hence use calculus as a tool: one can use strategies such as "the inverse function of this function satisfies a certain differential equation, which implies it can't be elementary". But here you have to be skillful to actually get your equation into the right form. This is why I can't use Nicholas' answer to answer my question. Nicholas observes that Lambert's $W$ function isn't elementary, and I know a proof of this which uses calculus and Liouville's criterion -- the trick being to write down a differential equation which (a simple function of) $W$ satisfies and then using Liouville to prove that this differential equation has no elementary solutions -- but the moment you change the problem a little, the differential equation changes, and the methods may not (and in this case don't) apply. An analogue might be this: if I can integrate $1/\sqrt{1-x^2}$, then even though it looks quite similar, I might well not be able to integrate $1/\sqrt{1-x^2+x^{-2}}$. This is an area where even quite a small change might derail things substantially.

On the other hand, these answers helped me immensely, because they provided me with references which enabled me to start a literature search. And today the search came to an end, because I got my hands on a copy of "Integration in finite terms: Liouville's theory of elementary methods" by J. F. Ritt (written in 1948), and on p56 Ritt shows how to deal with $x+\sin(x)$ explicitly! Apparently the equation $y=x+a\sin(x)$ ($a$ a constant) is called "Kepler's equation".

So the answer to the question is "this has nothing really to do with differential Galois theory; you need Liouville's theory, and a proof is in p56 of Ritt's 1948 book mentioned above".