TITLE: How to check if some equation can be solved using Lambert $\operatorname{W}$ function.
QUESTION [13 upvotes]: I'm very interested in Lambert $\operatorname{W}$ function and I want to know how to check if some equation can be solved using this function.
Example $1$:
$$e^xx=a$$
For this equation it is obviously that $x=\operatorname{W}_k(a)$ where $k\in\mathbb{Z}$.
Example $2$:
$$a^xx=b$$
Now we must reduce it to form $e^{f(x)}f(x)=c$ and then use Lambert $\operatorname{W}$ function.
$$e^{x\ln a}x=b$$
$$e^{x\ln a}x\ln a=b\ln a$$
$$x\ln a=\operatorname{W}_k(b\ln a)$$
$$x=\dfrac{\operatorname{W}_k(b\ln a)}{\ln a}$$
This is not too hard to solve.
Example $3$:
$$a^x+bx+c=0$$
It is very hard to solve this and after long computation we will get
$$x=\dfrac{-b\operatorname{W}_k\left(\dfrac{\ln a\cdot a^{-\dfrac{c}{b}}}{b}\right)-c\ln a}{b\ln a}$$
Example $4$:
$$a^{x^2}+bx+c=0$$
There is no known solution for this equation.
Example $5$:
$$\sin x+x=a$$
or
$$\dfrac i 2e^{-ix}-\dfrac i 2e^{ix}+x=a$$
My question is how to check if some equation can be solved using Lambert $\operatorname{W}$ function.

REPLY [4 votes]: Assume an ordinary equation $F(x)=c$ is given where $c$ is a constant and $F$ is a function. Isolating $x$ on one side of the equation only by operations to the whole equation means to apply a suitable partial inverse function (branch of the inverse relation) $F^{-1}$ of $F$: $\ x=F^{-1}(c)$.
The problem of existence of elementary inverses of elementary functions is solved by the theorem in [Ritt 1925] that is proved also in [Risch 1979].
The problem of existence of elementary numbers as solutions of irreducible polynomial equations $P(x,e^x)=0$ is solved in [Lin 1983] and [Chow 1999].
But LambertW is not an elementary function.
I. a. the following kinds of equations of $x$ can be solved in closed form by applying Lambert W or without Lambert W.
Let
$c_1,...,c_8\in\mathbb{C}$,
$f,f_1$ functions in $\mathbb{C}$ with suitable local closed-form inverses.
$$\tag 1 c_1x^{c_2}+c_3x^{c_4}\left(e^{c_5+c_6x^{c_7}}\right)^{c_8}=0$$
$$\tag 2 c_1f(x)^{c_2}+c_3f(x)^{c_4}\left(e^{c_5+c_6f(x)^{c_7}}\right)^{c_8}=0$$
If $c_2,c_4\neq0$, $x=0$ and $f(x)=0$ respectively is a solution.
$$\tag 3 c_1+c_2x+e^{c_3+c_4x}=0$$
$$\tag 4 f_1\left(c_1+c_2f(x)+e^{c_3+c_4f(x)}\right)=0$$
If you have an $x$-containing summand on the left-hand side of the equation, you can subtract the exponential term from the equation and divide the equation by it to get a product of a power of $x$ (or $f(x)$) and an exponential term of $x$ (or $f(x)$) to apply Lambert W.
See also [Edwards 2020], [Galidakis/Weisstein], [Köhler].
If your equation contains an exponential function or a logarithm function but cannot be brought to the form of equations (1) - (4), you could try to apply a predefined generalization of Lambert W.
See e.g. [Corcino/Corcino/Mezö 2017], [Dubinov/Galidakis 2007], [Galidakis 2005], [Maignan/Scott 2016], [Mezö 2017], [Barsan 2018].
$\ $
[Barsan 2018] Barsan, V.: Siewert solutions of transcendental equations, generalized Lambert functions and physical applications. Open Phys. 16 (2018) 232–242
[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
[Corcino/Corcino/Mezö 2017] Corcino, C. B.; Corcino, R. B.; Mezö, I.: Integrals and derivatives connected to the r-Lambert function. Integral Transforms and Special Functions 28 (2017) (11)
[Dubinov/Galidakis 2007] Dubinov, A.; Galidakis, Y.: Explicit solution of the Kepler equation. Physics of Particles and Nuclei Letters 4 (2007) 213-216
[Edwards 2020] Edwards, S.: Extension of Algebraic Solutions Using the Lambert W Function. 2020
[Galidakis 2005] Galidakis , I. N.: On solving the p-th complex auxiliary equation $f^{(p)}(z)=z$. Complex Variables 50 (2005) (13) 977-997
[Galidakis/Weisstein] Galidakis, I.; Weisstein, E. W.: Power Tower. Wolfram MathWorld
[Köhler] Köhler, Th: Gebrauch der Lambertschen W-Funktion (Omegafunktion)
[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
[Maignan/Scott 2016] Maignan, A.; Scott, T. C.: Polynomial-Exponential and Generalized Lambert Function. 2016
[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934
[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