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.
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1$\begingroup$ I immediately upvoted your question simply for mentioning the possibility of solving the sine equation via the Lambert function (Example 5). Just last week it occurred to me to wonder if the magic of complex numbers allows for a trigonometric analogue, but I couldn't find any info on the subject. $\endgroup$– David HCommented Dec 2, 2014 at 11:28
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$\begingroup$ I have plenty of questions/answers that use this connection. It is rather amazing in my opinion. $\endgroup$– Simply Beautiful ArtCommented Dec 19, 2015 at 18:12
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$\begingroup$ @DavidH Did you try checking the "trigonometry" tag? I've solved plenty there. $\endgroup$– Simply Beautiful ArtCommented Dec 19, 2015 at 18:21
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$\begingroup$ I have found you can solve some solutions of iterated exponential functions, as noted in my post math.stackexchange.com/questions/1583907/… . You can solve $e^{e^{e^x}}=x$ with my methods! :D $\endgroup$– Simply Beautiful ArtCommented Dec 21, 2015 at 0:41
2 Answers
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], [Mezö/Baricz 2017], [Barsan 2018], [Castle 2018].
$\ $
[Castle 2018] Castle, P.: Taylor series for generali-zed Lambert W functions. 2018
[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)
[Edwards 2020] Edwards, S.: Extension of Algebraic Solutions Using the Lambert W Function. 2020
[Galidakis/Weisstein] Galidakis, I.; Weisstein, E. W.: Power Tower. Wolfram MathWorld
[Köhler] Köhler, Th: Gebrauch der Lambertschen W-Funktion (Omegafunktion)
[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equa-tion. J. Math. Anal. Appl. 455 (2017) (1) 538-553
[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)
Generally, you can't solve something in the following form:$$x^{x+c}$$because of the stupid "$c$".
If you can't get it into the form $f(x)e^{f(x)}$, it is also probably not solvable.
You usually can't solve $$f(x)g(x)$$where $f(x)$ is more than one "level" away from $g(x)$. For example:$$xe^{e^x}$$has no solution because $x$ is two "levels" away from $e^{e^x}$.
Negative levels can be treated as logarithms.
The Lambert W function does not like addition, which is why you usually cancel addition by exponentiation and applying exponent properties.
Any base is considered fine (usually) because you can change the base with exponent/logarithm properties as well.
