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Consider the problem of solving for y in the following equations:

$$e^y+y=x \\ \sin y\ + y = x $$

I have often heard it said that "these types of problems" (finding the inverse of a function with a mixture of polynomial and transcendental functions) are, in general, impossible to do analytically.

I am wondering is there a way to prove this?

Edit (clarification): One of the comments mentioned that the use of the word "transcendental" in the question is sufficient as to imply the impossibility of solving for y. This may well be the case however what I am looking for is a proof that this is the case. For example: Prove that one can't find an function $y=f(x)$ which is both elementary and satisfies $e^y+y=x$.

Isaac Greene
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  • Something from https://math.stackexchange.com/questions/46934/what-is-the-solution-of-cosx-x might provight insight. I still struggle with the fact that for $\cos x = x, x$ has no "closed form," i.e. can't be written without summation or helping functions (that's how I defined 'closed') – Christopher Marley Jan 25 '19 at 05:51
  • I think that the word transcendental by itself says everything https://en.wikipedia.org/wiki/Transcendental_equation Some special functions can be used in some cases, such as the first equation you wrote; the solution of it being $y=x-W\left(e^x\right)$ where appears Lambert function. – Claude Leibovici Jan 25 '19 at 06:06

1 Answers1

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$$e^y+y=x\tag{1}$$ $$\sin(y)+y=x\tag{2}$$

Equation (2) becomes

$$-\frac{1}{2}\left(e^{iy}-e^{-iy}\right)+y=x.$$

1.) Series

If you allow series as analytic solutions, your equations can be solved by Lagrange inversion. Sometimes, an explicit formula for the general term of a series can be found.

Let's consider closed-form solutions.

2.) Elementary functions

Your equations are equations of elementary functions. You are looking for the inverses (means the inverse functions) of the functions $y\mapsto e^y+y$ and $y\mapsto\sin(y)+y$ respectively over domains where the functions are bijective.

Equation (1) can be solved in terms of Lambert W, equation (2) is Kepler's equation.

Your equations cannot be solved in terms of elementary functions (means by elementary inverses). This is directly proven e.g. in [Liouville 1837, 1838], [Ritt 1948] page 56, [Rosenlicht 1969], [Bronstein/Corless/Davenport/Jeffrey 2008].

The main theorem in [Ritt 1925], that's proven also in [Risch 1979], suggests that the elementary functions above cannot have elementary inverses. The main theorems in [Lin 1983] and [Chow 1999] imply, assuming Schanuel's conjecture is true, that your equations cannot have solutions that are elementary numbers or explicit elementary numbers if $x$ is an elementary number. This implies, assuming Schanuel's conjecture is true, that the elementary functions above over non-discrete domains cannot have elementary inverses therefore.

3.) Lambert W

Equation (1) is solved by $y=x-W_k(e^x)$. But as indicated above, the corresponding function is not an elementary function.
$\ $

[Bronstein/Corless/Davenport/Jeffrey 2008] 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

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

[Liouville 1837, 1838] 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

[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759

[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90

[Ritt 1948] Ritt, J. F.: Integration in finite terms. Liouville's theory of elementary methods. Columbia University Press, New York, 1948

[Rosenlicht 1969] Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22

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