I’m very interested in Lambert W function and I want to know how to check if some equation can be solved using this function.

Example 1:

exx=a

For this equation it is obviously that x=Wk(a) where k∈Z.Example 2:

axx=b

Now we must reduce it to form ef(x)f(x)=c and then use Lambert W function.

exlnax=b

exlnaxlna=blna

xlna=Wk(blna)

x=Wk(blna)lna

This is not too hard to solve.Example 3:

ax+bx+c=0

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

x=−bWk(lna⋅a−cbb)−clnablna

Example 4:

ax2+bx+c=0

There is no known solution for this equation.Example 5:

sinx+x=a

or

i2e−ix−i2eix+x=a

My question ishow to check if some equation can be solved using Lambert W function.

**Answer**

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,ex)=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

c1,...,c8∈C,

f,f1 functions in C with suitable local closed-form inverses.

c1xc2+c3xc4(ec5+c6xc7)c8=0

c1f(x)c2+c3f(x)c4(ec5+c6f(x)c7)c8=0

If c2,c4≠0, x=0 and f(x)=0 respectively is a solution.

c1+c2x+ec3+c4x=0

f1(c1+c2f(x)+ec3+c4f(x))=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].

[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)

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