# How to check if some equation can be solved using Lambert W\operatorname{W} function.

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 $1$:

For this equation it is obviously that $x=\operatorname{W}_k(a)$ where $k\in\mathbb{Z}$.

Example $2$:

Now we must reduce it to form $e^{f(x)}f(x)=c$ and then use Lambert $\operatorname{W}$ function.

This is not too hard to solve.

Example $3$:

It is very hard to solve this and after long computation we will get

Example $4$:

There is no known solution for this equation.

Example $5$:

or

My question is how to check if some equation can be solved using Lambert $\operatorname{W}$ function.

Assume an ordinary equation $$F(x)=cF(x)=c$$ is given where $$cc$$ is a constant and $$FF$$ is a function. Isolating $$xx$$ 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) $$F−1F^{-1}$$ of $$FF$$: $$x=F−1(c)\ x=F^{-1}(c)$$.

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 $$P(x,ex)=0P(x,e^x)=0$$ is solved in [Lin 1983] and [Chow 1999].

But LambertW is not an elementary function.

I. a. the following kinds of equations of $$xx$$ can be solved in closed form by applying Lambert W or without Lambert W.

Let
$$c1,...,c8∈Cc_1,...,c_8\in\mathbb{C}$$,
$$f,f1f,f_1$$ functions in $$C\mathbb{C}$$ with suitable local closed-form inverses.

$$c1xc2+c3xc4(ec5+c6xc7)c8=0\tag 1 c_1x^{c_2}+c_3x^{c_4}\left(e^{c_5+c_6x^{c_7}}\right)^{c_8}=0$$

$$c1f(x)c2+c3f(x)c4(ec5+c6f(x)c7)c8=0\tag 2 c_1f(x)^{c_2}+c_3f(x)^{c_4}\left(e^{c_5+c_6f(x)^{c_7}}\right)^{c_8}=0$$

If $$c2,c4≠0c_2,c_4\neq0$$, $$x=0x=0$$ and $$f(x)=0f(x)=0$$ respectively is a solution.

$$c1+c2x+ec3+c4x=0\tag 3 c_1+c_2x+e^{c_3+c_4x}=0$$

$$f1(c1+c2f(x)+ec3+c4f(x))=0\tag 4 f_1\left(c_1+c_2f(x)+e^{c_3+c_4f(x)}\right)=0$$

If you have an $$xx$$-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 $$xx$$ (or $$f(x)f(x)$$) and an exponential term of $$xx$$ (or $$f(x)f(x)$$) to apply Lambert W.

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 $$f(p)(z)=zf^{(p)}(z)=z$$. 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