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

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 kZ.

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(lnaacbb)clnablna
Example 4:
ax2+bx+c=0
There is no known solution for this equation.

Example 5:
sinx+x=a
or
i2eixi2eix+x=a
My question is how 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) F1 of F:  x=F1(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,...,c8C,
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,c40, 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].
 

[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

Attribution
Source : Link , Question Author : Community , Answer Author : IV_

Leave a Comment